\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 29, pp. 1--25.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/29\hfil Construction of almost periodic functions]
{Construction of almost periodic functions with given properties}

\author[M. Vesel\'y\hfil EJDE-2011/29\hfilneg]
{Michal Vesel\'y}

\address{Michal Vesel\'y \newline
Department of Mathematics and Statistics\\
Masaryk University\\
Kotl\'a\v{r}sk\'a 2 \\
CZ-611 37 Brno, Czech Republic}
\email{michal.vesely@mail.muni.cz}

\thanks{Submitted August 22, 2010. Published February 18, 2011.}
\thanks{Supported by grant 201/09/J009 from the Czech Grant Agency}
\subjclass[2000]{34A30, 34C27, 42A75}
\keywords{Almost periodic solutions; linear differential equations;
\hfill\break\indent skew-Hermitian systems}

\begin{abstract}
We define almost periodic functions with values in a pseudometric
space $\mathcal{X}$. We mention the Bohr and the Bochner
definition of almost periodicity. We present one modifiable method
for constructing almost periodic functions in $\mathcal{X}$.
Applying this method, we prove that in any neighbourhood of an
almost periodic skew-Hermitian linear differential system there
exists a system which does not possess a nontrivial almost
periodic solution.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks


\section{Introduction}

This paper is motivated by \cite{ves0000p} where almost periodic
\emph{sequences} and linear  \emph{difference} systems are
considered. Here we will consider almost periodic \emph{functions}
and linear \emph{differential} systems. Our aim is to show a way
one can generate almost periodic functions with several prescribed
properties. Since our process can be used for generalizations of
classical (complex valued) almost periodic functions, we introduce
the almost periodicity in pseudometric spaces and we present our
method for almost periodic functions with values in a pseudometric
space~$\mathcal{X}$.

Note that we obtain the most important case if $\mathcal{X}$  is a
Banach space, and that the theory of almost periodic functions of
real variable with values in a Banach space, given by  Bochner
\cite{ves797bo}, is in its essential lines similar to the theory
of classical almost periodic functions which is due to Bohr
\cite{vesbohr1}, \cite{vesbohr2}. We introduce almost periodic
functions in pseudometric spaces using a trivial extension of the
Bohr concept, where the modulus is replaced by the distance. In
the classical case, we refer to the monographs 
\cite{ves711co,ves798fi}; for functions with values in Banach spaces,
to \cite{ves795ap}, \cite[Chapter VI]{ves711co}; for other
extensions, to \cite{vesfunako, vessharma}.

Necessary and sufficient conditions for a continuous function with
values in a Banach space to be almost periodic may be no longer
valid for continuous functions in general metric spaces. For the
approximation condition, it is seen that the completeness of the
space of values is necessary and Tornehave \cite{ves794to} also
requires the local connection by arcs of the space of values. In
the Bochner condition, it suffices to replace the convergence by
the Cauchy condition. Since we need the Bochner concept as well,
we recall that the Bochner condition means that any sequence of
translates of a given continuous function has a subsequence which
converges, uniformly on the domain of the function. The fact, that
this condition is equivalent with the Bohr definition of almost
periodicity in Banach spaces, was proved by Bochner
\cite{ves797bo}.

The above mentioned Bohr definition and Bochner condition are
formulated in Section 2 (with some basic properties of almost
periodic functions). In this section, processes from
\cite{ves711co} are generalized. Analogously, the theory of almost
periodic functions of real variable with fuzzy real numbers as
values is developed in~\cite{vesbede1}. We remark that fuzzy real
numbers form a complete metric space.

In Section 3, we mention the way one can construct almost periodic
functions with prescribed properties in a pseudometric space. We
present it in Theorems \ref{th3.v}, \ref{th32}, \ref{th33}. Note
that it is possible to obtain many modifications and
generalizations of our process. A special construction of almost
periodic functions with given properties is published (and
applied) in~\cite{vesjaza}.

Then we will analyse almost periodic solutions of almost periodic
linear differential systems. Sometimes this field is called the
Favard theory what is based on the Favard contributions in
\cite{vesfava} (see also \cite[Theorem~1.2]{vesaut2},
\cite[Theorem~6.3]{ves798fi} or \cite[Theorem 1]{vesorttar}; for
homogeneous case, see \cite{vescoppel}, \cite{vesfavard}). In this
context, sufficient conditions for the existence of almost
periodic solutions are mentioned in \cite{veshumin} (for
generalizations, see \cite{veshumin2}, \cite{vesishii}; for other
extensions and supplements of the Favard theorem, see
\cite{ves797bo}, \cite{vesaut2}, \cite{ves798fi} and the
references cited therein). Certain sufficient conditions, under
which \emph{homogeneous} systems that have nontrivial bounded
solutions also have nontrivial almost periodic solutions, are
given in \cite{ves4PAL}.

It is a corollary of the Favard and the Floquet theory that any
bounded solution of an almost periodic linear differential system
is almost periodic if the matrix valued function, which determines
the system, is periodic (see \cite[Corollary 6.5]{ves798fi}; for a
generalization in the homogeneous case, see \cite{veshara}). This
result is no longer valid for systems with almost periodic
coefficients. There exist systems for which all solutions are
bounded, but none of them is almost periodic (see, e.g.,
\cite{vesjohnr}, \cite{vessellgr}). Homogeneous systems have the
zero solution which is almost periodic, but do not need to have an
other almost periodic solution. We note that the existence of a
homogeneous system, which has bounded solutions (separated from
zero) and, at the same time, all systems from some neighbourhood
of it do not have any nontrivial almost periodic solution, is
proved in~\cite{ves704tk}.

We will consider the set of  systems of the form
\begin{equation} \label{ves64op3}
x' = A (t)  x,
\end{equation}
where $A$ is almost periodic and all matrices $A(t)$, $t \in
\mathbb{R}$, are skew-Hermitian, with the uniform topology of
matrix functions $A$ on the real axis. In \cite{ves705tk}, it is
proved that the systems \eqref{ves64op3}, all of whose solutions
are almost periodic, form a dense subset of the set of all
considered systems. We add that special cases of this result are
proved in \cite{ves70kur,ves71kur}. For systems whose solutions
are not almost periodic, we refer to~\cite{ves020th}.

In Section 4, using the method for constructing almost periodic
functions from Section 3, we will prove that, in any neighbourhood
of a system of the form~\eqref{ves64op3}, there exists a system
which does not possess an almost periodic solution other than the
trivial one, not only with a fundamental matrix which is not
almost periodic as in \cite{ves020th}. It means that, applying our
method, we will get a stronger version of a statement
from~\cite{ves020th}. We remark that constructions of almost
periodic homogeneous linear differential system with given
properties are used in \cite{veslipa,veslipn} as well.

Let $\mathcal{X}$ be an arbitrary pseudometric space with a
pseudometric $\varrho$; i.e., let
$$
\varrho(x, x) = 0, \quad \varrho(x, y) = \varrho(y, x) \ge 0,
 \quad \varrho(x, y) \le \varrho(x,z)
+ \varrho(z, y)
$$
for any $x, y, z \in \mathcal{X}$. The symbol
$\mathcal{O}_\varepsilon (x)$ will denote the
$\varepsilon$-neighbourhood of $x$ in $\mathcal{X}$ for arbitrary
$\varepsilon >0$, $x \in \mathcal{X}$ and $\mathbb{R}^{+}_0$ the
set of all nonnegative real numbers. For the used notations, we
can also refer to \cite{ves0000p}.

\section{Almost periodic functions in pseudometric spaces}

We introduce the Bohr almost periodicity in $\mathcal{X}$.
Observe that we are not able to
distinguish between
$x \in \mathcal{X}$ and $y \in \mathcal{X}$ if $\varrho (x, y) = 0$.

\begin{definition} \label{def1} \rm
A continuous function $\psi: \mathbb{R} \to \mathcal{X}$
is \emph{almost periodic} if for any $\varepsilon > 0$, there
exists a number $p(\varepsilon)> 0$ with the property that any
interval of length $p(\varepsilon)$ of the real line contains at
least one point $s$, such that
$$
\varrho ( \psi ({t + s}), \psi (t) ) < \varepsilon, \quad t \in \mathbb{R}.
$$
The number $s$ is called an $\varepsilon$-\emph{translation number}
and the set
of all ${\varepsilon}$-translation numbers of $\psi$
is denoted by~$\mathfrak{T} (\psi, \varepsilon)$.
\end{definition}

If $\mathcal{X}$ is a Banach space, then a continuous function
$\psi$ is almost periodic if and only if any set of translates of
$\psi$ has a subsequence, uniformly convergent on $\mathbb{R}$ in
the sense of the norm. See, e.g., \cite[Theorem 6.6]{ves711co}.
Evidently, this result cannot be longer valid if the space of
values is not complete. Nevertheless, we prove the below given
Theorem \ref{thm21}, where the convergence is replaced by the
Cauchy condition. Before proving this statement, we mention two
simple lemmas. Their proofs can be easily obtained by modifying
the proofs of \cite[Theorem 6.2]{ves711co} and
\cite[Theorem 6.5]{ves711co}, respectively.

\begin{lemma} \label{lemma11}
An almost periodic function with values in $\mathcal{X}$
is uniformly continuous on the real line.
\end{lemma}

\begin{lemma} \label{lemma12} The set
of all values of an almost periodic function
$\psi:\mathbb{R} \to \mathcal{X}$
is totally bounded in~$\mathcal{X}$.
\end{lemma}

Now we can formulate the main result of this section,
which we will apply in Section 4.

\begin{theorem} \label{thm21}
Let $\psi: \mathbb{R} \to \mathcal{X}$ be a continuous function.
Then, $\psi$ is almost periodic if and only if from any sequence
of the form $\{ \psi (t + {s_{n}})\}_{n \in \mathbb{N}}
$, where $s_n$ are real numbers, one can extract a subsequence
$\{ \psi (t + {{r}_{n}})\}_{n \in \mathbb{N}} $ 
satisfying the Cauchy uniform convergence condition on
$\mathbb{R}$; i.e., for any $\varepsilon > 0$, there exists $ l
(\varepsilon) \in \mathbb{N}$ with the property that
$$
\varrho (\psi (t + r_{i}), \psi (t + r_{j})) < \varepsilon , \quad t \in \mathbb{R}
$$
for all $i, j > l(\varepsilon)$, $i, j \in \mathbb{N}$.
\end{theorem}

\begin{proof}
The sufficiency of the condition can
be proved using a simple extension of the argument used in the
proof of \cite[Theorem 1.10]{ves711co}. In that proof, it is
only supposed, by contradiction, that any sequence of translates
of $\psi$ has a subsequence which satisfies the Cauchy uniform
convergence condition, and that $\psi$ is not almost periodic.
Thus, it suffices to replace the modulus by the distance in the
proof of \cite[Theorem 1.10]{ves711co}.

To prove the converse implication, we will assume
that $\psi$ is an almost periodic function,
and we will apply the well-known method of diagonal extraction
and modify the proof of \cite[Theorem 6.6]{ves711co}.

Let $\{ {t_{n}}; n \in \mathbb{N}\}$ be a dense
subset of $\mathbb{R}$ and $\{ {s_{n}} \}_{n \in \mathbb{N}} 
\subset \mathbb{R}$ be an arbitrarily given sequence.
From the sequence $ \{ \psi (t_1 + {{s}_{n}}) \}_{n \in \mathbb{N}}$,
using Lemma \ref{lemma12}, we choose a subsequence $ \{ \psi
(t_1 + {{r}_{n}^1}) \}_{n \in \mathbb{N}}$ such that, for any
$\varepsilon > 0$, there exists $ l_1 (\varepsilon) \in \mathbb{N}$
with the property that
$$
\varrho (\psi (t_1 + {{r}_{i}^1}), \psi (t_1 + {{r}_{j}^1} ))
< \varepsilon, \quad i, j > l_1 (\varepsilon), \, i, j \in \mathbb{N}.
$$
Such a subsequence exists because infinitely many values
$\psi (t_1 + {{s}_{n}})$ is in a neigh\-bourhood of radius~$2^{-i}$
for all $i \in \mathbb{N}$ (consider the method of diagonal
extraction). Analogously, from the sequence
$ \{ \psi (t_2 + {{r}_{n}^1}) \}_{n \in \mathbb{N}}$, we obtain
$ \{ \psi (t_2 + {{r}_{n}^2}) \}_{n \in \mathbb{N}}$ such that, for any
$\varepsilon > 0$, there exists $ l_2 (\varepsilon) \in \mathbb{N}$
for which
$$
\varrho (\psi (t_2 + {{r}_{i}^2}), \psi (t_2 + {{r}_{j}^2})
) < \varepsilon, \quad i, j > l_2 (\varepsilon), \, i, j
\in \mathbb{N}.
$$
We proceed further in the same way. We obtain
$\{{r}_{n}^k\} \subseteq \dots \subseteq \{{r}_{n}^1\} $,
$k \in \mathbb{N}$.

Let $\varepsilon > 0$ be arbitrarily given, $p =
p({\varepsilon}/{5})$ be from Definition \ref{def1},
$\delta = \delta ({\varepsilon}/{5})$ correspond to
${\varepsilon}/{5}$ from the definition of the uniform continuity
of $\psi$ (see Lemma \ref{lemma11}) and let a finite set
$\{{t_{1}}, \dots, {t_{j}} \} \subset \{ {t_{n}}; {n \in \mathbb{N}} \}$ 
satisfy
$$
\min _{i \in \{1, \dots, j\}} | t_{i} - t | < \delta, \quad
t \in [0, p].
$$
Obviously, there exists $l \in \mathbb{N}$
such that, for all integers $n_1, n_2 > l$, it is valid
$$
\varrho (\psi (t_{i} + {r}_{n_1}^{n_1}), \psi (t_{i}
+ {r}_{n_2}^{n_2} )) <  \frac{\varepsilon}{5}, \quad
i \in \{1, \dots, j\}.
$$
Let $t \in \mathbb{R}$ be given, $s = s(t) \in [-t, -t + p]$ be
an ${\varepsilon}/{5}$-translation number of $\psi$, and
$t_i = t_i (s) \in \{{t_{1}}, \dots, {t_{j}} \}$ be such that
$|t + s - t_i | < \delta$.
Finally, we have
\begin{align*}
&\varrho (\psi (t + r_{n_1}^{n_1}),
 \psi (t + {r}_{n_2}^{n_2})) \\
&\le  \varrho (\psi (t + {r}_{n_1}^{n_1}),
 \psi (t + {r}_{n_1}^{n_1} + s ))
+  \varrho (\psi (t + {{r}_{n_1}^{n_1}} + s),
 \psi (t_i + {r}_{n_1}^{n_1} )) \\
& + \varrho (\psi (t_i + {{r}_{n_1}^{n_1}}),
 \psi (t_i + {r}_{n_2}^{n_2}))
+ \varrho (\psi (t_i + {{r}_{n_2}^{n_2}}),
  \psi (t + {r}_{n_2}^{n_2} + s )) \\
& + \varrho (\psi (t + {{r}_{n_2}^{n_2}} + s),
 \psi (t + {r}_{n_2}^{n_2} )).
\end{align*}
Thus, we obtain
\begin{equation} \label{ves1j3x}
\varrho (\psi (t + {{r}_{n_1}^{n_1}}),
\psi (t + {r}_{n_2}^{n_2} ))
 < \frac{\varepsilon}{5} + \frac{\varepsilon}{5}
 + \frac{\varepsilon}{5}+ \frac{\varepsilon}{5}
+ \frac{\varepsilon}{5} = {\varepsilon}
\end{equation}
for all $t \in \mathbb{R}$, $n_1, n_2 > l$, $n_1, n_2 \in \mathbb{N}$.
Evidently, \eqref{ves1j3x}
completes the proof of the theorem if we put
${r}_{n} := {r}_{n}^{n}$, ${n \in \mathbb{N}}$.
\end{proof}

Analogously as for almost periodic functions with values in a
Banach space, one can prove many properties of almost periodic
functions in a pseudometric space. For example, the limit of a
uniformly convergent sequence of almost periodic functions is
almost periodic (see \cite[Theorem 6.4]{ves711co}).

Using Theorem \ref{thm21} $n$-times,
one can also obtain:

\begin{corollary} \label{cor1}
If $\mathcal{X}_1, \dots, X_n$
are pseudometric spaces and $\psi_1, \dots , \psi_n$
are arbitrary almost periodic functions with values in 
$\mathcal{X}_1, \dots, \mathcal{X}_n$,
respectively, then the function~$ \psi$, with values
in $ \mathcal{X}_1 \times \dots \times \mathcal{X}_n$
given by $ \psi := (\varphi_1, \dots, \varphi_n)$,
is almost periodic as well.
\end{corollary}

Moreover, from Corollary \ref{cor1} it follows that
the set
$$
\mathfrak{T} (\psi_1, \varepsilon) \cap \mathfrak{T}
(\psi_2, \varepsilon) \cap \dots  \cap \mathfrak{T}
(\psi_n, \varepsilon)
$$
is relative dense in $\mathbb{R}$ for arbitrary almost periodic
functions $\psi_1, \psi_2, \dots , \psi_n $ and any $\varepsilon>0$.
We add that one can use Corollary \ref{cor1} to obtain simple
modifications of the below presented method of constructions of
almost periodic functions.

\section{Construction of almost periodic functions}

Now we present the way
one can generate almost periodic functions with given properties
in the next theorem.

\begin{theorem} \label{th3.v} For arbitrary $a>0$,
any continuous function $\psi: \mathbb{R} \to \mathcal{X}$
such that
\begin{gather*}
 \psi (t) \in \mathcal{O}_{a} (\psi (t-1)), \quad t \in (1, 2], \\
 \psi (t) \in \mathcal{O}_{a} (\psi (t + 2)), \quad t \in (-2, 0], \\
 \psi (t) \in \mathcal{O}_{a/2} (\psi (t - 4)), \quad t \in (2, 6],\\
 \psi (t) \in \mathcal{O}_{a/2} (\psi (t + 8)), \quad t \in (-10, -2],\\
 \psi (t) \in \mathcal{O}_{a/4} (\psi (t - 2^4)), \quad
 t \in (2 + 2^2, 2 + 2^2 + 2^4],\\
 \psi (t) \in \mathcal{O}_{a/4} (\psi (t + 2^5)), \quad
 t \in (-2^5 - 2^3 -2, - 2^3 -2],\\
\dots \\
 \psi (t) \in \mathcal{O}_{a\,2^{-n}} (\psi (t - 2^{2n})), \quad
t \in (2 + 2^2 + \dots + 2^{2n - 2}, 2 + 2^2 + \dots + 2^{2n - 2}
 + 2^{2n}], \\
 \psi (t) \in \mathcal{O}_{a\,2^{-n}} (\psi (t + 2^{2n + 1})), \quad
t \in (-2^{2n + 1} - \dots - 2^3 -2, -2^{2n - 1} - \dots - 2^3 -2], \\
 \dots
\end{gather*}
is almost periodic.
\end{theorem}

\begin{proof}
 Let $\varepsilon>0$ be arbitrary
and $k = k (\varepsilon) \in \mathbb{N}$ be such that
$ {2^k} > 8 a/\varepsilon$.
We have to prove that
the set of all $\varepsilon$-translation numbers of $\psi$
is relative dense in~$\mathbb{R}$.
We will obtain this from the fact that $ l \cdot 2^{2k} $
is an $\varepsilon$-translation number of $\psi$ for any integer~$l$.
We see that
 \begin{gather*}
\psi (t) \in \mathcal{O}_{\varepsilon / 8} (\psi (t - 2^{2k})), \quad
t \in (2 + 2^2 + \dots + 2^{2k - 2}, 2 + 2^2 + \dots + 2^{2k}], \\
\psi (t) \in \mathcal{O}_{\varepsilon / 8} (\psi (t + 2^{2k + 1})), \,\,
t \in (-2^{2k + 1} - \dots - 2^3 -2, -2^{2k - 1} - \dots - 2^3 -2], \\
\psi (t) \in \mathcal{O}_{\varepsilon / 16} (\psi (t - 2^{2k + 2})),
 \quad t \in (2 + 2^2 + \dots + 2^{2k}, 2 + 2^2 + \dots + 2^{2k + 2}],\\
\dots
\end{gather*}
In a pseudometric space $\mathcal{X}$, this implies
\begin{gather*}
\psi (t + 2^{2k}) \in \mathcal{O}_{\varepsilon / 8} (\psi (t)), \quad
 t \in [-2^{2k - 1} - \dots - 2^3 -2, 2 + 2^2 + \dots + 2^{2k - 2}], \\
\psi (t - 2^{2k + 1}) \in \mathcal{O}_{\varepsilon / 8} (\psi (t)), \quad
 t \in [-2^{2k - 1} - \dots - 2^3 -2, 2 + 2^2 + \dots + 2^{2k - 2}], \\
\psi (t - 2^{2k}) \in \mathcal{O}_{\varepsilon / 8 + \varepsilon / 8}
 (\psi (t)), \quad
 t \in [-2^{2k - 1} - \dots - 2^3 -2, 2 + 2^2 + \dots + 2^{2k - 2}], \\
\psi (t + 2^{2k + 1}) \in \mathcal{O}_{\varepsilon/8
  + \varepsilon/16} (\psi (t)), \quad
 t \in [-2^{2k - 1} - \dots - 2^3 -2, 2 + 2^2 + \dots + 2^{2k - 2}], \\
\begin{aligned}
&\psi (t + 3 \cdot 2^{2k}) \in \mathcal{O}_{\varepsilon/8
 + \varepsilon/8 +  \varepsilon/16} (\psi (t)), \\
 &t \in [-2^{2k - 1} - \dots - 2^3 -2, 2 + 2^2 + \dots + 2^{2k - 2}],
\end{aligned}\\
\psi (t + 2^{2k + 2}) \in \mathcal{O}_{\varepsilon/16} (\psi (t)), \quad
 t \in [-2^{2k - 1} - \dots - 2^3 -2, 2 + 2^2 + \dots + 2^{2k - 2}], \\
\begin{aligned}
&\psi (t + 2^{2k} + 2^{2k + 2}) \in \mathcal{O}_{\varepsilon/8
 + \varepsilon/16} (\psi (t)), \\
 &t \in [-2^{2k - 1} - \dots - 2^3 -2, 2 + 2^2 + \dots + 2^{2k - 2}],
\end{aligned}\\
\dots
 \end{gather*}
Since
$$
\frac{\varepsilon}{8} + \frac{\varepsilon}{8}
+ \frac{\varepsilon}{16} + \frac{\varepsilon}{16}
+ \frac{\varepsilon}{32} + \dots = \frac{\varepsilon}{2},
$$
we have
\begin{equation} \label{vestt60b}
\begin{aligned}
&\psi (t + l \cdot 2^{2k}) \in \mathcal{O}_{\varepsilon/2} (\psi (t)),\\
&t \in [-2^{2k - 1} - \dots - 2^3 -2, 2 + 2^2 + \dots + 2^{2k - 2}],
\quad l \in \mathbb{Z}.
\end{aligned}
\end{equation}
We express any $t \in \mathbb{R}$ as the sum of numbers $p(t)$ and
$q(t)$ for which
\begin{gather*}
p (t) \in [-2^{2k - 1} - \dots - 2^3 -2, 2 + 2^2 + \dots + 2^{2k - 2}], \\
q (t) \in \mathbb{Z} \quad \text{and} \quad
q(t) = j 2^{2k} \quad \text{for some } j \in \mathbb{Z}.
\end{gather*}
Using \eqref{vestt60b}, we obtain
\begin{equation}  \label{veskt831}
\begin{aligned}
&\varrho (\psi (t), \psi ({t + l\cdot 2^{2k}}))\\
&\le \varrho (\psi (p(t) + q(t)), \psi (p(t)))
+ \varrho (\psi (p(t)), \psi ({p(t) + (j + l) \,2^{2k}}))
< \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon
\end{aligned}
\end{equation}
for any $t \in \mathbb{R}$, $l \in \mathbb{Z}$, which completes
 the proof.
\end{proof}

The process mentioned in the previous theorem is easily modifiable.
We illustrate this fact by the following two theorems. We remark that
Theorem \ref{th32} is used in \cite{vesvesa}.

\begin{theorem} \label{th32}
Let $ M > 0 $, $x_0 \in \mathcal{X}$,
and $j \in \mathbb{N}$ be given.
Let $\varphi: [ 0,  M ] \to \mathcal{X}$ satisfy
$\varphi (0) = \varphi (M) = x_0$.
If $\{r_{n}\}_{n \in \mathbb{N}} \subset \mathbb{R}_0^+$
has the property that
\begin{equation} \label{vess11a}
\sum_{n=1}^\infty { r_{n}} < \infty,
\end{equation}
then an arbitrary continuous function
$\psi: \mathbb{R} \to \mathcal{X}$,
$\psi \arrowvert_{[ 0,  M ]} \equiv \varphi$
for which
\begin{align*}
\psi (t) = x_0, \quad
 t &\in \{i M, \, 2 \le i \le j + 1\} \cup
 \{- i (j+1) M, \, 1 \le i \le j \}  \cup \\
&\cup_{n = 1}^{\infty} {\{((j +1) + \dots + j(j + 1)^{2n-2}
 + i (j + 1)^{2n})M; \, 1 \le i \le j \}}  \cup \\
&\cup_{n = 1}^{\infty} {\{-((j +1) + \dots + j(j + 1)^{2n-1}
 + i (j + 1)^{2n + 1})M; \, 1 \le i \le j \}}
\end{align*}
and, at the same time, for which it is valid
\begin{gather*}
 \psi (t) \in \mathcal{O}_{r_{1}} (\psi (t-M)), \quad t \in (M, 2 M), \\
 \dots \\
 \psi (t) \in \mathcal{O}_{r_{1}} (\psi (t - jM)), \quad t \in
 (jM , (j+1) M),  \\
 \psi (t) \in \mathcal{O}_{r_{2}} (\psi (t + (j+1)M)), \quad
 t \in (-(j+1)M , 0), \\
 \dots \\
 \psi (t) \in \mathcal{O}_{r_{2}} (\psi (t + j(j+1)M)), \quad
 t \in (-j(j+1)M , - (j-1)(j +1) M),\\
 \psi (t) \in \mathcal{O}_{r_{3}} (\psi (t - (j+1)^2 M)), \quad
 t \in ((j+1) M, ((j+1) + (j+1)^2 )M), \\
 \dots \\
 \begin{aligned}
&\psi (t) \in \mathcal{O}_{r_{3}} (\psi (t - j(j +1)^2 M)),\\
&t \in (((j +1) + (j - 1)(j+1)^2)M, ((j +1) + j(j +1)^2)M),
\end{aligned} \\
\dots \\
\begin{aligned}
&\psi (t) \in \mathcal{O}_{r_{2n}} (\psi (t + (j + 1)^{2n - 1} M)), \\
&t \in \bigl(-((j + 1)^{2n - 1} + j(j + 1)^{2n - 3} + \dots
 + j(j +1)^3 + j(j +1)) M, \\
& \quad -(j(j + 1)^{2n - 3} + \dots + j(j +1)^3 + j(j +1))M\bigr),
\end{aligned} \\
\dots \\
\begin{aligned}
&\psi (t) \in \mathcal{O}_{r_{2n}} (\psi (t + j(j + 1)^{2n - 1} M)), \\
&t \in \bigl(-(j(j + 1)^{2n - 1}+ j(j + 1)^{2n - 3} + \dots
 + j(j +1)^3 + j(j +1)) M, \\
 &  \quad -((j-1)(j + 1)^{2n - 1} + j(j + 1)^{2n - 3}
 + \dots + j(j +1)^3 + j(j +1))M\bigr),
\end{aligned} \\
\begin{aligned}
&\psi (t) \in \mathcal{O}_{r_{2n + 1}} (\psi (t - (j + 1)^{2n} M)), \\
&t \in \bigl(((j +1) + j(j +1)^2 +  \dots + j(j+1)^{2n - 2})M, \\
&  \quad ((j +1) + j(j +1)^2 + \dots + j(j+1)^{2n - 2}+ (j + 1)^{2n}) M \bigr),
\end{aligned} \\
 \dots \\
\begin{aligned}
&\psi (t) \in \mathcal{O}_{r_{2n + 1}} (\psi (t - j(j + 1)^{2n} M )), \\
&t \in \bigl(((j +1) + j(j +1)^2 + \dots + j(j+1)^{2n - 2}
 + (j-1)(j + 1)^{2n})M, \\
&  \quad ((j +1) + j(j +1)^2 + \dots + j(j+1)^{2n - 2}
 + j(j + 1)^{2n})M \bigr),
\end{aligned} \\
\dots
\end{gather*}
is almost periodic.
\end{theorem}

\begin{proof}
We can prove this theorem analogously as Theorem \ref{th3.v}.
Let $\varepsilon$ be a positive number and let an odd integer
$n(\varepsilon) \ge 2$
have the property (see \eqref{vess11a}) that
\begin{equation} \label{vesrkk1}
\sum_{n=n(\varepsilon)}^\infty { r_{n} } < \frac{\varepsilon}{2}.
\end{equation}
We will prove that $ l(j + 1)^{n(\varepsilon) - 1} M $
is an $\varepsilon$-translation number of $\psi$ for all
$l \in \mathbb{Z}$.
Arbitrarily choosing $l \in \mathbb{Z}$ and $t \in \mathbb{R}$,
if we put
\begin{equation} \label{vesre11}
s:= l(j + 1)^{n(\varepsilon) - 1} M,
\end{equation}
then it suffices to show that the inequality
\begin{equation} \label{vesre9}
d (\psi (t), \psi (t + s)) < \varepsilon
\end{equation}
holds; i.e., this inequality proves the theorem.

We can write $t$ as the sum of numbers $t_1$ and $t_2$, where
\begin{equation}  \label{vesre29}
\begin{gathered}
{t}_1 \ge -\bigl(j(j+1)^{n(\varepsilon) -2} + \dots + j(j+1)^3 +j(j+1) \bigr)M, \\
{t}_1 \le \bigl(j +1 + j(j +1)^2 + \dots + j(j + 1)^{n(\varepsilon) -3}\bigr) M
\end{gathered}
\end{equation}
and
\begin{equation} \label{vesre28}
t_2 = i (j+1)^{n(\varepsilon) -1} M \quad \text{for some }
i \in \mathbb{Z}.
\end{equation}
Now we have (see \eqref{vesre29} and the proof of Theorem \ref{th3.v})
\begin{equation}  \label{vesj00b}
\begin{aligned}
\varrho (\psi (t),\psi (t + s))
&\le \varrho (\psi(t_1 + t_2), \psi (t_1))
 + \varrho (\psi(t_1), \psi (t_1 + t_2 + s))\\
&< \sum_{n=n(\varepsilon)}^{n(\varepsilon) + p - 1} { r_{n}}
 + \sum_{n=n(\varepsilon)}^{n(\varepsilon) +  q - 1 } { r_{n}}.
\end{aligned}
\end{equation}
Indeed, we can express (consider \eqref{vesre11} and \eqref{vesre28})
\begin{align*}
t_2  &= \bigl(i_1 (j+1)^{n(\varepsilon) -1}
 + i_2 (j+1)^{n(\varepsilon)} + \dots + i_p (j+1)^{n(\varepsilon)
 + p - 1} \bigr) (m + 1), \\
t_2 + s &= \bigl(l_1 (j+1)^{n(\varepsilon) -1}
+ l_2 (j+1)^{n(\varepsilon)} + \dots + l_q (j+1)^{n(\varepsilon)
+ q - 1} \bigr) (m + 1),
\end{align*}
where $i_1, \dots, i_p, l_1, \dots, l_q \subseteq \{-j, \dots, 0, \dots, j \}$
satisfy
$$
i_1 \ge 0,  i_2 \le 0, \dots,  (-1)^p \, i_p \le 0,
\quad l_1 \ge 0, l_2 \le 0, \dots,  (-1)^q \, l_q \le 0,
 $$
and use (iii).
It is sure that \eqref{vesrkk1} and \eqref{vesj00b}
give \eqref{vesre9}.
\end{proof}

\begin{theorem} \label{th33}
Let $\varphi : (-r,  r ] \to \mathcal{X}$,
$\{r_{n}\}_{n \in \mathbb{N}} \subset \mathbb{R}^{+}_0$,
and $\{j_{n}\}_{n \in \mathbb{N}} \subseteq \mathbb{N}$
be arbitrary such that
\begin{equation} \label{veso5l}
\sum_{n=1}^\infty { r_{n} j_n} < \infty
\end{equation}
holds, and let a function $\psi: \mathbb{R} \to \mathcal{X}$ satisfy
$\psi \arrowvert_{(-r, r]} \equiv \varphi$ and
\begin{gather*}
 \psi (t) \in \mathcal{O}_{r_{1}} (\varphi (t - 2r)) , \quad
 t \in (r , r + 2r ],\\
\dots  \\
 \psi (t) \in \mathcal{O}_{r_{1}} (\varphi ({t - 2r})) , \quad
 t \in (r+ (j_1 - 1)2r , r + j_1 2r ], \\
\psi (t) \in \mathcal{O}_{r_{1}} (\varphi (t +2r)), \quad
 t \in (-2r -r , -r ], \\
 \dots  \\
\psi (t) \in \mathcal{O}_{r_{1}} (\varphi (t + 2r)) , \quad
 t \in (-j_1 2r -r , -(j_1 -1) 2r- r ], \\
 \dots \\
\psi (t) \in \mathcal{O}_{r_{n}} (\varphi (t - p_n)) , \quad
t \in (p_1 + \dots + p_{n-1} , p_1 + \dots + p_{n-1} + p_n ], \\
 \dots  \\
\begin{aligned}
&\psi (t) \in \mathcal{O}_{r_{n}} (\varphi (t - p_n)), \\
&t \in (p_1 +  \dots + p_{n-1} + (j_{n} - 1)p_n , p_1 +  \dots
 + p_{n-1} + j_{n} p_n],
\end{aligned} \\
 \psi (t) \in \mathcal{O}_{r_{n}} (\varphi (t + p_n)), \quad
 t \in (- p_{n} - p_{n-1} - \dots - p_1 , - p_{n-1} - \dots - p_1 ],\\
\dots  \\
\begin{aligned}
& \psi (t) \in \mathcal{O}_{r_{n}} (\varphi (t + p_n)), \\
& t \in (- j_n p_n - p_{n-1} - \dots - p_1 , - (j_{n} - 1) p_{n}
- p_{n-1} - \dots - p_1 ],
\end{aligned} \\
\dots
\end{gather*}
where
\begin{gather*}
 p_1:= r + j_1 2r, \quad p_2 := 2(r + j_1 2r), \\
 p_3 := (2 j_2 + 1) p_2, \quad \dots, \quad
 p_n := (2j_{n-1} + 1)p_{n-1}, \dots
\end{gather*}
If $\psi$ is continuous on $\mathbb{R}$, then it is almost periodic.
\end{theorem}

\begin{proof}
 It is not difficult to prove Theorem \ref{th33} analogously as
Theorems \ref{th3.v} and \ref{th32}. For given $\varepsilon > 0$,
let an integer $n(\varepsilon) \ge 2$ satisfy
$$
\sum_{n=n(\varepsilon)}^\infty { r_{n}  j_n } < \frac{\varepsilon}{4}.
$$
One can prove the inclusion
\begin{equation} \label{vesolx9}
 \{ l p_{n(\varepsilon)};  l \in \mathbb{Z} \} \subseteq
\mathfrak{T} (\psi, \varepsilon)
\end{equation}
which yields the almost periodicity of $\psi$.
 \end{proof}

\begin{remark} \label{rmk2} \rm
From the proofs of Theorems \ref{th3.v}, \ref{th32}, \ref{th33}
(see \eqref{veskt831}, \eqref{vesre11} and \eqref{vesre9},
\eqref{vesolx9}), we obtain a property of the set of all
${\varepsilon}$-translation numbers of the resulting
function~$\psi$. For any ${\varepsilon} > 0$, there exists nonzero
$c \in \mathbb{R}$ for which
$$
\{ l c;  l \in \mathbb{Z} \} \subseteq \mathfrak{T}
(\psi, \varepsilon).
$$
An important class of almost periodic functions is the
class of \emph{limit-periodic} functions. To this class belong
the uniform limits of sequences of periodic continuous functions
(in general, having different periods).
It is seen that, applying the method from the above
theo\-rems, we obtain
li\-mit-pe\-riodic functions.
\end{remark}

\section{An application}

Let $m \in \mathbb{N}$ be arbitrarily given. In this section, we
will use the following notations: $\mathcal{I}m ({\varphi})$ for
the range of a function $\varphi$, $\mathcal{M}at (\mathbb{C}, m)$
for the set of all $m \times m$ matrices with complex elements,
$U (m) \subset \mathcal{M}at (\mathbb{C}, m) $ for the group of all
unitary matrices of dimension~$m$, $ A^*$ for the conjugate
transpose of $A \in \mathcal{M}at (\mathbb{C}, m)$, $I$ for the
identity matrix, $0$ for the zero matrix, and $i$ for the
imaginary unit.

We will analyse systems of $m$ homogeneous linear
differential equations of the form
\begin{equation} \label{ves697i}
x' (t) = A (t)  x (t), \quad  t \in \mathbb{R},
\end{equation}
where $A$ is an almost periodic function with
$\mathcal{I}m (A) \subset \mathcal{M}at (\mathbb{C}, m)$
and with the property that
$A (t) + A^* (t) = 0$ for any $t \in \mathbb{R}$;
i.e., $A: \mathbb{R} \to \mathcal{M}at (\mathbb{C}, m)$
is an almost periodic function of skew-Hermitian (skew-adjoint)
matrices.
Let $\mathcal{S}$ be
\emph{the set of all systems} \eqref{ves697i}.
We will identify the function $A$ with the system \eqref{ves697i}
which is determined by~$A$. Especially,
we will write $A \in \mathcal{S}$.

In the vector space $\mathbb{C}^m$, we will consider the absolute
norm $\| \cdot \|_1$ (one can also
consider the Euclidean norm or the maximum norm). Let
$\|  \cdot  \|$ be the corresponding
matrix norm. Considering that every almost periodic function is
bounded (see Lemma \ref{lemma12}), the distance between two
systems $A, B \in \mathcal{S}$ is defined by the norm of the matrix
valued functions $A$, $B$, uniformly on~$\mathbb{R}$; i.e., we
introduce the metric
$$
\sigma (A , B) := \sup_{t \in \mathbb{R}}
\| A(t) - B(t) \|, \quad A, B \in \mathcal{S}.
$$
For $\varepsilon >0$, the symbol $\mathcal{O}_\varepsilon^\sigma (A)$
will denote the $\varepsilon$-\emph{neighbourhood} of $A$ in
$\mathcal{S}$.

Now we recall the notion of the frequency module and its rational
hull which can be introduced for all almost periodic function with
values in a Banach space. \emph{The frequency
module} $\mathcal{F}$ of an almost periodic function $A:\mathbb{R}
\to \mathcal{M}at (\mathbb{C}, m)$ is the $\mathbb{Z}$-module of the
real numbers, generated by the $\lambda$ such that
$$
\lim_{T \to +\infty}  \frac{1}{T}  \int_0^T {A(t)
 e^{2\pi i \lambda t}\, dt} \ne 0.
$$
\emph{The rational hull of}~$\mathcal{F}$ is the set
$$
 \{\lambda/l;  \lambda \in \mathcal{F},  l \in \mathbb{Z}\}.
$$
For the frequency modules of almost periodic linear differential
systems and their solutions, we refer to
 \cite[Chapters~4, 6]{ves798fi},
\cite{vesorttar}.

In \cite{ves705tk}, it is proved that, in any neighbourhood of a
system \eqref{ves697i} with frequency module $\mathcal{F}$,
there exists a system with a frequency module contained in the
rational hull of $\mathcal{F}$ possessing all almost periodic
solutions with frequencies belonging to the rational hull of
$\mathcal{F}$ as well. From \cite[Theorem~1]{ves4TK5} it follows
that there exists a system \eqref{ves697i} which cannot be
approximated by the so-called reducible systems with frequency
module~$\mathcal{F}$ (there exists an open set of irreducible
systems with a fixed frequency module); i.e., a neighbourhood of a
system \eqref{ves697i} may not contain a system with almost
periodic solutions and frequency module~$\mathcal{F}$. In this
case, see also \cite{ves4ELI} and \cite{ves4Tuv} for reducible
constant systems and systems reducing to diagonal form by Lyapunov
transformation with frequency module~$\mathcal{F}$, respectively.

In addition, it is proved in \cite{ves020th} that the systems with
$k$-dimensional frequency basis of $A$, having solutions which
are not almost periodic, form a subset of the second category of
the space of all considered systems with $k$-dimen\-sional
frequency basis of $A$. Thus, it is known (see
\cite[Corollary 1]{ves705tk}) that the systems with
$k$-dimen\-sional frequency basis of $A$ and with an almost
periodic fundamental matrix form a dense set of the first category
in the space of all systems~\eqref{ves697i} with
$k$-dimen\-sional frequency basis.

In this context, we formulate the following result that the
systems having no nontrivial almost periodic solution form a dense
subset of $\mathcal{S}$.

\begin{theorem} \label{th4.1}
For any $A \in \mathcal{S}$ and $\varepsilon > 0$, there exists
$B \in \mathcal{O}_\varepsilon^\sigma (A)$ which
does not have an almost periodic solution other than the trivial one.
\end{theorem}

\begin{proof}
Let $A, C \in \mathcal{S}$ and $\varepsilon > 0$ be arbitrary.
Since the sum of skew-Hermitian matrices is also skew-Hermitian
and since the sum of two almost periodic functions is almost
pe\-riodic (consider Theorem \ref{thm21}), we have that $A + C \in
\mathcal{S}$. Let $X_A (t)$, $t \in \mathbb{R}$ and
$X_C (t)$, $t \in \mathbb{R}$ be the principal
(i.e., $X_A (0) = X_C (0) = I$)
fundamental matrices of $A \in \mathcal{S}$ and $C \in \mathcal{S}$,
respectively. If the matrices $C(t)$, $X_A(t)$ commute for all
$t \in \mathbb{R}$, then the matrix valued function
$X_A (t) \cdot X_C (t)$, $t \in \mathbb{R}$ is the principal
fundamental matrix of $A + C \in \mathcal{S}$. Indeed,
from $X'_A (t) = A (t) \cdot X_A (t)$,
$X'_C (t) = C (t) \cdot X_C (t)$ for $t \in \mathbb{R}$, we
obtain
\begin{align*}
(X_A (t)  X_C (t) )'
&= A(t) X_A (t)  X_C (t) +  X_A (t)  C(t)  X_C (t) \\
&= A(t) X_A (t)  X_C (t) + C(t) X_A (t) X_C (t)\\
&= (A + C)(t) X_A (t)  X_C (t), \quad t \in \mathbb{R}.
\end{align*}
It gives that it suffices to find
$C \in \mathcal{O}_\varepsilon^\sigma (0)$
for which all matrices $C(t)$, $t \in \mathbb{R}$ have the form
$ \operatorname{diag} [ia, \dots, ia ]$, $a \in \mathbb{R}$
and for which the vector valued function
$X_A (t) \cdot X_C (t) \cdot u$, $t \in \mathbb{R}$ is not almost
periodic for any vector $ u \in \mathbb{C}^m$, $\|u \|_1 = 1$.

We will construct such an almost periodic function $C$ applying
Theorem \ref{th3.v}
for $a = \varepsilon/4$. First of all we put
$$
C(t)\equiv 0, \quad t \in [0, 1].
$$
Then, in the first step of our construction, we define $C$
on $(1, 2]$ arbitrarily so that it is
constant on $[1 +1/4, 1 + 3/4]$ and $\| C(t)\| < \varepsilon/4$
for $t$ from this interval, $C(2):= C(1) = 0$, and it is linear
between values $0$, $C(3/2)$ on $[1, 1 + 1/4]$
and $[1 + 3/4, 2]$.

In the second step, we define continuous $C$
satisfying $\|C(t) - C(t + 2) \| < \varepsilon/4$
for $t \in [-2, 0)$
arbitrarily so that it is constant on
$$
[-2 + 1/16, -2 + 1 - 1/16], \quad
[-2 + 1 + 1/4 + 1/16, -2 + 1 +  3/4 - 1/16];
$$
at the same time, we put
\begin{gather*}
 C(-2) := C(0) = 0, \quad C(-1 + 1/4) := C(1 + 1/4) = C(3/2),\\
 C(-1) := C(1) = 0, \quad C(-1/4) := C(2 - 1/4) = C(3/2),\\
 C(t) \equiv C(3/2)/2, \quad
t \in [-1 + 1/16, -1 +1/4 - 1/16] \cup [-1/4 + 1/16, - 1/16]
\end{gather*}
and  define $C$ so that it is linear on
\begin{gather*}
[-2, -2 + 1/16], \quad [-1 - 1/16, -1], \quad [-1, -1 + 1/16],\\
[-1 +1/4 - 1/16, -1 + 1/4], \quad [-1 + 1/4, -1 +1/4 + 1/16], \\
[-1/4 - 1/16, -1/4], \quad [-1/4, -1/4 + 1/16], \quad [-1/16, 0].
\end{gather*}
Analogously, in the third step, we obtain $C$ on $(2, 6]$ for which
we can choose constant values on
\begin{gather*}
[4 -2 + 1/16 + 8^{-1}/16, 4 -2 + 1 - 1/16 - 8^{-1}/16],\\
[4 -2 + 1 + 1/4 + 1/16 + 8^{-1}/16, 4 -2 + 1 + 3/4 - 1/16 - 8^{-1}/16],\\
[4 -1 + 1/16 + 8^{-1}/16, 4 -1 +1/4 - 1/16 - 8^{-1}/16],\\
[4 -1/4 + 1/16 + 8^{-1}/16, 4 - 1/16 - 8^{-1}/16],\\
[4 + 8^{-1}/16, 4 + 1 - 8^{-1}/16], \quad
[4 + 1 + 1/4 + 8^{-1}/16, 4 + 1 + 3/4 - 8^{-1}/16]
\end{gather*}
arbitrarily so that
$\|C(t) - C(t - 4) \| < \varepsilon/8$,
$t \in (2, 6]$; at the same time, we put
\begin{gather*}
C(4 - 2 + 1/16) := C(-2 + 1/16) = C(-3/2),\\
C(4 - 2 + 1 - 1/16) := C(-1 - 1/16) = C(-3/2),\\
C(4 - 1):= C(-1) = 0,\\
C(4 - 1 + 1/16) := C(- 1 + 1/16) = C(3/2)/2,\\
C(4 - 1 +1/4 - 1/16) := C(- 1 +1/4 - 1/16) = C(3/2)/2,\\
C(4 -1 + 1/4) := C(-1 + 1/4) = C(3/2),\\
C(4 -2 +1 + 1/4 + 1/16) := C (-2 +1 + 1/4 + 1/16) = C(-1/2),\\
C(4 -2 + 1 + 3/4 - 1/16) := C(-2 + 1 + 3/4 - 1/16) = C(-1/2),\\
C(4 -1/4) := C(-1/4) = C(3/2),\\
C(4 -1/4 + 1/16) := C(-1/4 + 1/16) = C(3/2)/2,\\
C(4 -1/16) := C(- 1/16) = C(3/2)/2,\\
C(4) := C(0), \quad C(4 + 1) := C (1),\\
C(4 + 1 + 1/4) := C(1 + 1/4) = C(3/2),\\
C(4 + 1 + 3/4) := C(1 + 3/4) = C(3/2),\\
C(4 + 2) := C (2) = C (0) = 0,\\
\begin{aligned}
C(t) \equiv C(-3/2)/2, \quad  t &\in [4 -2 + 8^{-1}/16, 4 -2 + 1/16 - 8^{-1}/16] \\
&\cup [4 - 1 - 1/16 + 8^{-1}/16, 4 - 1 - 8^{-1}/16],
\end{aligned}\\
\begin{aligned}
C(t) \equiv C(3/2)/4, \quad t &\in [4 -1 + 8^{-1}/16, 4 - 1 + 1/16 - 8^{-1}/16] \\
&\cup [4 -1/16 + 8^{-1}/16, 4 - 8^{-1}/16],
\end{aligned}\\
\begin{aligned}
C(t) \equiv 3\,C(3/2)/4, \quad t &\in [4 - 1 +1/4 - 1/16 + 8^{-1}/16,
4 -1 +1/4 - 8^{-1}/16] \\
& \cup [4 - 1/4 + 8^{-1}/16, 4 - 1/4 +1/16 - 8^{-1}/16],
\end{aligned}\\
\begin{aligned}
C(t) \equiv &(C(3/2) + C(-1/2))/2, \\
t &\in [4 - 1 +1/4 + 8^{-1}/16,
4 -1 +1/4 + 1/16 - 8^{-1}/16] \\
&\cup [4 - 1/4 -1/16 + 8^{-1}/16, 4 - 1/4 - 8^{-1}/16],
\end{aligned}\\
 \begin{aligned}
C(t) \equiv &(8\,C(4 + 1) + 1\,C(4 + 1 + 1/4))/9, \\
&t \in [4 + 1 + 8^{-1}/16, 4 + 1 + 8^{-1}/16 \cdot 3],
\end{aligned}\\
\begin{aligned}
C(t) \equiv \,&(7\,C(4 + 1) + 2\,C(4 + 1 + 1/4))/9, \\
&t \in [4 + 1 + 8^{-1}/16 \cdot 5, 4 + 1 + 8^{-1}/16 \cdot 7],
\end{aligned}\\
\dots \\
\begin{aligned}
C(t) \equiv &(1\,C(4 + 1) + 8\,C(4 + 1 + 1/4))/9, \\
&t \in [4 + 1 + 8^{-1}/16 \cdot 29, 4 + 1 + 8^{-1}/16 \cdot 31],
\end{aligned}\\
\begin{aligned}
C(t) \equiv &(8\,C(4 + 1 + 3/4) + 1\,C(4 + 2))/9, \\
&t \in [4 + 1 + 3/4 + 8^{-1}/16, 4 + 1 + 3/4 + 8^{-1}/16 \cdot 3],
 \end{aligned}\\
\begin{aligned}
C(t) \equiv \,&(7\,C(4 + 1 + 3/4) + 2\,C(4 + 2))/9, \\
&t \in [4 + 1 + 3/4 + 8^{-1}/16 \cdot 5, 4 + 1 + 3/4
 + 8^{-1}/16 \cdot 7],
\end{aligned}\\
 \dots \\
\begin{aligned}
C(t) \equiv \,&(1\, C(4 + 1 + 3/4) + 8\,C(4 + 2))/9, \\
&t \in[4 + 1 + 3/4 + 8^{-1}/16 \cdot 29, 4 + 1 + 3/4
 + 8^{-1}/16 \cdot 31]\,.
\end{aligned}
\end{gather*}
Then we define continuous $C$ so that it is linear on the
rest of subintervals.

If we denote
\begin{gather*}
a_1^1 := 0, \quad b_1^1 := 0, \quad c_1^1 := 1, \\
a_2^1 := 1, \quad b_2^1 := 1 + 1/4, \quad c_2^1 := 1 + 3/4,
 \quad a_3^1 := 2
\end{gather*}
and (compare with the situation after the second step)
\begin{gather*}
a_1^2 := -2, \quad b^2_1 := -2, \quad c_1^2  := -2,\\
a_2^2 := -2, \quad b_2^2 := -2 + 1/16, \quad c^2_2 := - 1 - 1/16, \\
a_3^2 := - 1, \quad b^2_3 := -1, \quad c^2_3:= -1, \\
a_4^2 := -1, \quad b^2_4 := -1 + 1/16, \quad c^2_4:= -1 + 1/4 - 1/16,\\
a_5^2 := - 1 + 1/4, \quad b_5^2 := -1 + 1/4 + 1/16, \quad
 c_5^2 := -1 + 3/4 - 1/16, \\
a_6^2 := -1 + 3/4, \quad b_6^2 := -1 + 3/4 + 1/16, \quad
 c_6^2 := - 1/16,
\end{gather*}
we see that $C$ does not need to be constant only on
\begin{gather*}
[a_j^1 - 2, a_j^1 - 2 + 4^{-2}], \quad
[b_2^1 -2 - 4^{-2}, b_2^1 -2], \quad
[b_j^1 - 2, b_j^1 -2 + 4^{-2}], \\
[c_j^1 -2 - 4^{-2}, c_j^1 - 2], \quad
[c_j^1 -2, c_j^1 -2 + 4^{-2}], \quad
[a_{j+1}^1 - 2 - 4^{-2}, a_{j+1}^1 - 2]
\end{gather*}
for $j \in \{1, 2\}$; i.e., on
$$
[a_j^2, b_j^2], \, j \in \{1, \dots, 6\}, \quad [c_j^2, a_{j+1}^2], \,
 j \in \{1, \dots, 5\}, \quad [c_6^2, 0], 
$$
and it has to be constant on each one of the intervals
$$[a_2^1 - 2 + 4^{-2}, b_2^1 - 2 - 4^{-2}], \quad [c_2^1 - 2 + 4^{-2}, a_{3}^1 - 2 - 4^{-2}],$$
$$[b_j^1 -2 + 4^{-2}, c_j^1 - 2 - 4^{-2}], \quad j \in \{1, 2\}, $$
i.e., on
$[b_j^2, c_j^2]$, $j \in \{1, \dots, 6\}$.
It is also seen that
$$
a_1^2 = d_1^1, \quad b_1^2 = d_2^1, \quad  c_1^2 = d_3^1, \quad
 a_2^2 = d_4^1, \quad \dots \quad  c_{6}^2 = d_{18}^1,
$$
where $d_1^1, d_2^1, \dots, d^1_{18}$ is the nondecreasing
sequence of all numbers
\begin{gather*}
a_j^1 - 2, \quad b_j^1 - 2, \quad c_j^1 - 2, \\
\min \{ a_j^1 - 2 + 4^{-2}, b_j^1 - 2 \}, \quad
 \max \{ a_j^1 - 2, b_j^1 - 2 - 4^{-2} \}, \\
\min \{ c_j^1 - 2, b_j^1 - 2 + 4^{-2} \}, \quad
\max \{ c_j^1 - 2 - 4^{-2}, b_j^1 - 2\} , \\
\min \{ c_j^1 - 2 + 4^{-2}, a_{j+1}^1 - 2\}, \quad
\max \{ c_j^1 - 2, a_{j+1}^1 - 2 - 4^{-2} \}
\end{gather*}
for $j \in \{1, 2\}$. We put $a_{7}^2 := 0$.

Let $d_1^2, d_2^2, \dots, d^2_{168}$ be the nondecreasing sequence of
all numbers
\begin{gather*}
b_1^1  + 4, \quad b_1^1  + 4, \quad b_1^1  + 4, \quad
 b_1^1  + 4, \quad b_1^1  + 4, \quad b_1^1  + 4, \\
b_1^1  + 4, \quad b_1^1  + 4, \quad b_1^1  + 4, \quad
 b_1^1  + 4, \quad b_1^1  + 4, \quad b_1^1  + 4, \\
c_1^1  + 4, \quad \min \{ c_1^1 + 4, b_1^1 + 4 + 8^{-1}/16 \}, \quad
 \max \{ c_1^1 + 4 - 8^{-1}/16, b_1^1 + 4\}, \\
c_2^1  + 4, \quad \min \{ c_2^1 + 4, b_2^1 + 4 + 8^{-1}/16 \}, \quad
 \max \{ c_2^1 + 4 - 8^{-1}/16, b_2^1 + 4\}, \\
a_1^1 + (4k + 1)(b_1^1 - a_1^1)/32 + 4, \quad
 a_1^1 + (4k + 3)(b_1^1 - a_1^1)/32 + 4, \\
a_1^1 + (4k + 4)(b_1^1 - a_1^1)/32 + 4, \quad
 k \in \{0, 1, \dots, 7\},\\
c_1^1 + (4k + 1)(a_2^1 - c_1^1)/32 + 4, \quad
 c_1^1 + (4k + 3)(a_2^1 - c_1^1)/32 + 4, \\
c_1^1 + (4k + 4)(a_2^1 - c_1^1)/32 + 4, \quad
 k \in \{0, 1, \dots, 7\}, \\
a_2^1 + (4k + 1)(b_2^1 - a_2^1)/32 + 4, \quad
 a_2^1 + (4k + 3)(b_2^1 - a_2^1)/32 + 4, \\
a_2^1 + (4k + 4)(b_2^1 - a_2^1)/32 + 4, \quad
 k \in \{0, 1, \dots, 7\},\\
c_2^1 + (4k + 1)(a_3^1 - c_2^1)/32 + 4, \quad
 c_2^1 + (4k + 3)(a_3^1 - c_2^1)/32 + 4, \\
c_2^1 + (4k + 4)(a_3^1 - c_2^1)/32 + 4, \quad
 k \in \{0, 1, \dots, 7\}
\end{gather*}
and
\begin{gather*}
 a_{j + 1}^2 + 4, \quad b_j^2 + 4, \quad c_j^2 + 4, \\
\min \{ a_j^2 + 4 + 8^{-1}/16, b_j^2 + 4 \}, \quad \max \{ a_j^2
 + 4, b_j^2  + 4 - 8^{-1}/16 \}, \\
\min \{ c_j^2  + 4, b_j^2  + 4 + 8^{-1}/16 \}, \quad
 \max \{ c_j^2  + 4 - 8^{-1}/16, b_j^2  + 4\}, \\
 \min \{ c_j^2  + 4 + 8^{-1}/16, a_{j+1}^2  + 4\}, \quad
 \max \{ c_j^2  + 4, a_{j+1}^2  + 4 - 8^{-1}/16 \}
\end{gather*}
for $j \in \{1, \dots, 6\}$. We denote
\[
a_{1}^3 := 2, \quad b_1^3 := d_1^2, \quad
c_1^3 := d_2^2, \quad a_2^3 := d_3^2, \quad \dots \quad
a_{57}^3 := d_{168}^2.
\]
 We remark that, in the sequences of $d_j^l$, $l \in \mathbb{N}$,
values are a number of time.

In the fourth step, we define $C$ so that
$$
\|C(t) - C(t + 2^{3}) \| < \frac{\varepsilon}{2^{3}}, \quad
t \in [- 2^3 -2, -2).
$$
We consider the nondecreasing sequence
$d_1^{3}, d_2^{3}, \dots,  d_{21 \cdot 8^2}^{3}$ of values
\begin{gather*}
a_j^3 - 2^3, \quad b_j^3 - 2^3, \quad c_j^3 - 2^3, \\
\min \{ a_j^3 - 2^3 + 8^{-2}/16, b_j^3 - 2^3 \}, \quad
 \max \{ a_j^3 - 2^3, b_j^3 - 2^3 - 8^{-2}/16 \}, \\
\min \{ c_j^3 - 2^3, b_j^3 - 2^3 + 8^{-2}/16 \}, \quad
\max \{ c_j^3 - 2^3 - 8^{-2}/16, b_j^3 - 2^3\}, \\
\min \{ c_j^3 - 2^3 + 8^{-2}/16, a_{j+1}^3 - 2^3\}, \quad
\max \{ c_j^3 - 2^3, a_{j+1}^3 - 2^3 - 8^{-2}/16 \}
\end{gather*}
for $j \in \{1, \dots, 7 \cdot 8\}$, $144$ numbers
$ b_1^1 - 2^3$, and
\begin{gather*}
 c_1^1  - 2^3, \quad \min \{ c_1^1 - 2^3, b_1^1 - 2^3 + 8^{-2}/16 \},
\quad \max \{ c_1^1 - 2^3 - 8^{-2}/16, b_1^1 - 2^3\}, \\
c_2^1  - 2^3, \quad \min \{ c_2^1 - 2^3, b_2^1 - 2^3 + 8^{-2}/16 \},
 \quad \max \{ c_2^1 - 2^3 - 8^{-2}/16, b_2^1 - 2^3\}, \\
\min \{ a_1^1 + (k-1)(b_1^1 - a_1^1)/(8 \cdot 4) - 2^3 + 8^{-2}/16,
 a_1^1 + k(b_1^1 - a_1^1)/(8 \cdot 4) - 2^3 \}, \\
 \max \{ a_1^1 + (k-1)(b_1^1 - a_1^1)/(8 \cdot 4) - 2^3,
 a_1^1 + k(b_1^1 - a_1^1)/(8 \cdot 4) - 2^3 - 8^{-2}/16 \}, \\
 a_1^1 + k (b_1^1 - a_1^1)/(8 \cdot 4) - 2^3, \quad
 k \in \{1, \dots, 8 \cdot 4\},\\
\min \{ c_1^1 + (k - 1)(a_2^1 - c_1^1)/(8 \cdot 4) - 2^3 + 8^{-2}/16,
 c_1^1 + k (a_2^1 - c_1^1)/(8 \cdot 4) - 2^3 \}, \\
\max \{ c_1^1 + (k - 1)(a_2^1 - c_1^1)/(8 \cdot 4) - 2^3,
 c_1^1 + k (a_2^1 - c_1^1)/(8 \cdot 4) - 2^3 - 8^{-2}/16 \}, \\
 c_1^1 + k (a_2^1 - c_1^1)/(8 \cdot 4) - 2^3, \quad k \in \{1, \dots, 8
 \cdot 4\},\\
\min \{ a_2^1 + (k-1)(b_2^1 - a_2^1)/(8 \cdot 4) - 2^3 + 8^{-2}/16,
 a_2^1 + k(b_2^1 - a_2^1)/(8 \cdot 4) - 2^3 \}, \\
\max \{ a_2^1 + (k-1)(b_2^1 - a_2^1)/(8 \cdot 4) - 2^3,
 a_2^1 + k(b_2^1 - a_2^1)/(8 \cdot 4) - 2^3 - 8^{-2}/16 \}, \\
a_2^1 + k (b_2^1 - a_2^1)/(8 \cdot 4) - 2^3, \quad
 k \in \{1, \dots, 8 \cdot 4\},\\
\min \{ c_2^1 + (k - 1)(a_3^1 - c_2^1)/(8 \cdot 4) - 2^3 + 8^{-2}/16,
 c_2^1 + k (a_3^1 - c_2^1)/(8 \cdot 4) - 2^3 \}, \\
\max \{ c_2^1 + (k - 1)(a_3^1 - c_2^1)/(8 \cdot 4) - 2^3,
  c_2^1 + k (a_3^1 - c_2^1)/(8 \cdot 4) - 2^3 - 8^{-2}/16 \}, \\
 c_2^1 + k (a_3^1 - c_2^1)/(8 \cdot 4) - 2^3, \quad k \in \{1, \dots,
 8 \cdot 4\},\\
 c_1^2 - 2^3, \quad \min \{ c_1^2 - 2^3, b_1^2 - 2^3 + 8^{-2}/16 \},
 \quad \max \{ c_1^2 - 2^3 - 8^{-2}/16, b_1^2 - 2^3\}, \\
c_2^2  - 2^3, \quad \min \{ c_2^2 - 2^3, b_2^2 - 2^3 + 8^{-2}/16 \},
 \quad \max \{ c_2^2 - 2^3 - 8^{-2}/16, b_2^2 - 2^3\}, \\
c_3^2 - 2^3, \quad \min \{ c_3^2 - 2^3, b_3^2 - 2^3 + 8^{-2}/16 \},
 \quad \max \{ c_3^2 - 2^3 - 8^{-2}/16, b_3^2 - 2^3\}, \\
c_4^2  - 2^3, \quad \min \{ c_4^2 - 2^3, b_4^2 - 2^3 + 8^{-2}/16 \},
 \quad \max \{ c_4^2 - 2^3 - 8^{-2}/16, b_4^2 - 2^3\}, \\
c_5^2 - 2^3, \quad \min \{ c_5^2 - 2^3, b_5^2 - 2^3 + 8^{-2}/16 \},
 \quad \max \{ c_5^2 - 2^3 - 8^{-2}/16, b_5^2 - 2^3\}, \\
 c_6^2  - 2^3, \quad \min \{ c_6^2 - 2^3, b_6^2 - 2^3 + 8^{-2}/16 \},
\quad \max \{ c_6^2 - 2^3 - 8^{-2}/16, b_6^2 - 2^3\}, \\
\min \{ a_1^2 + (k-1)(b_1^2 - a_1^2)/8 - 2^3 + 8^{-2}/16,
 a_1^2 + k(b_1^2 - a_1^2)/8 - 2^3 \}, \\
\max \{ a_1^2 + (k-1)(b_1^2 - a_1^2)/8 - 2^3,
 a_1^2 + k(b_1^2 - a_1^2)/8 - 2^3 - 8^{-2}/16 \}, \\
a_1^2 + k (b_1^2 - a_1^2)/8 - 2^3, \quad k \in \{1, \dots, 8\},\\
\min \{ c_1^2 + (k - 1)(a_2^2 - c_1^2)/8 - 2^3
+ 8^{-2}/16, c_1^2 + k (a_2^2 - c_1^2)/8 - 2^3 \}, \\
\max \{ c_1^2 + (k - 1)(a_2^2 - c_1^2)/8 - 2^3, c_1^2
+ k (a_2^2 - c_1^2)/8 - 2^3 - 8^{-2}/16 \}, \\
c_1^2 + k (a_2^2 - c_1^2)/8 - 2^3, \quad k \in \{1, \dots, 8\},\\
 \dots \\
\min \{ a_6^2 + (k-1)(b_6^2 - a_6^2)/8 - 2^3 + 8^{-2}/16, a_6^2
+ k(b_6^2 - a_6^2)/8 - 2^3 \}, \\
\max \{ a_6^2 + (k-1)(b_6^2 - a_6^2)/8 - 2^3,  a_6^2
 + k(b_6^2 - a_6^2)/8 - 2^3 - 8^{-2}/16 \}, \\
a_6^2 + k (b_6^2 - a_6^2)/8 - 2^3, \quad k \in \{1, \dots, 8\},\\
\min \{ c_6^2 + (k - 1)(a_7^2 - c_6^2)/8 - 2^3 + 8^{-2}/16, c_6^2
 + k (a_7^2 - c_6^2)/8 - 2^3 \}, \\
\max \{ c_6^2 + (k - 1)(a_7^2 - c_6^2)/8 - 2^3, c_6^2
 + k (a_7^2 - c_6^2)/8 - 2^3 - 8^{-2}/16 \}, \\
c_6^2 + k (a_7^2 - c_6^2)/8 - 2^3, \quad k \in \{1, \dots, 8\}.
\end{gather*}
We put
$$
a_{1}^4 := d_1^3, \quad b_1^4 := d_2^3, \quad c_1^4 := d_3^3,
\quad \dots \quad
c_{7 \cdot 8^2}^4 := d_{21 \cdot 8^2}^3,
\quad a_{7 \cdot 8^2 + 1}^4 := -2.
$$
We recall that $C$ can be increasing or decreasing only on
$$
[a^{4}_j, b^{4}_j], \quad [c^{4}_j, a^{4}_{j+1}], \quad
j \in \{1, \dots, 7 \cdot 8^2\}.
$$
We proceed further in the same way (as in the third and the fourth
step).
In the $2n$-th step, we define continuous $C$ so that
$$
\|C(t) - C(t + 2^{2n-1}) \| < \frac{\varepsilon}{2^{n+1}},
\quad t \in [-2^{2n - 1} - \dots -2,
-2^{2n - 3} - \dots -2).
$$
We get the nondecreasing sequence $\{d_l^{2n-1}\}$ from
\begin{gather*}
 a_j^{2n-1} - 2^{2n-1}, \quad b_j^{2n-1} - 2^{2n-1}, \quad
 c_j^{2n-1} - 2^{2n-1}, \\
 \min \{ a_j^{2n-1} - 2^{2n-1} + 8^{2 - 2n}/16, b_j^{2n-1}
 - 2^{2n-1} \}, \\
\max \{ a_j^{2n-1} - 2^{2n-1}, b_j^{2n-1} - 2^{2n-1}
 - 8^{2 - 2n}/16 \}, \\
 \min \{ c_j^{2n-1} - 2^{2n-1}, b_j^{2n-1} - 2^{2n-1}
 + 8^{2 - 2n}/16 \}, \\
\max \{ c_j^{2n-1} - 2^{2n-1} - 8^{2 - 2n}/16, b_j^{2n-1}
 - 2^{2n-1}\},\\
\min \{ c_j^{2n-1} - 2^{2n-1} + 8^{2 - 2n}/16, a_{j+1}^{2n-1}
 - 2^{2n-1}\}, \\
\max \{ c_j^{2n-1} - 2^{2n-1}, a_{j+1}^{2n-1} - 2^{2n-1}
- 8^{2 - 2n}/16 \}
\end{gather*}
for $j \in \{1, \dots, 7 \cdot 8^{2n -3}\}$, from
\begin{gather*}
 c_1^1  - 2^{2n-1}, \quad
 \min \{ c_1^1 - 2^{2n-1}, b_1^1 - 2^{2n-1} + 8^{2 - 2n}/16 \},\\
\max \{ c_1^1 - 2^{2n-1} - 8^{2 - 2n}/16, b_1^1 - 2^{2n-1}\}, \\
c_2^1  - 2^{2n-1}, \quad \min \{ c_2^1 - 2^{2n-1}, b_2^1 - 2^{2n-1}
+ 8^{2 - 2n}/16 \},\\
\max \{ c_2^1 - 2^{2n-1} - 8^{2 - 2n}/16, b_2^1 - 2^{2n-1}\}, \\
\begin{aligned}
\min \{ &a_1^1 + (k-1)(b_1^1 - a_1^1)/(8 \cdot 4^{2n - 3}) - 2^{2n-1}
 + 8^{2 - 2n}/16, \\
&a_1^1 + k(b_1^1 - a_1^1)/(8 \cdot 4^{2n - 3}) - 2^{2n-1} \},
\end{aligned}\\
\begin{aligned}
\max \{ &a_1^1 + (k-1)(b_1^1 - a_1^1)/(8 \cdot 4^{2n - 3}) - 2^{2n-1}, \\
&a_1^1 + k(b_1^1 - a_1^1)/(8 \cdot 4^{2n - 3}) - 2^{2n-1}
 - 8^{2 - 2n}/16 \},
\end{aligned}\\
 a_1^1 + k (b_1^1 - a_1^1)/(8 \cdot 4^{2n - 3}) - 2^{2n-1}, \quad
 k \in \{1, \dots, 8 \cdot 4^{2n - 3}\},\\
\begin{aligned}
 \min \{ &c_1^1 + (k - 1)(a_2^1 - c_1^1)/(8 \cdot 4^{2n - 3})
- 2^{2n-1} + 8^{2 - 2n}/16,\\
&c_1^1 + k (a_2^1 - c_1^1)/(8 \cdot 4^{2n - 3}) - 2^{2n-1} \},
\end{aligned} \\
\begin{aligned}
\max \{ &c_1^1 + (k - 1)(a_2^1 - c_1^1)/(8 \cdot 4^{2n - 3})
 - 2^{2n-1}, \\
&c_1^1 + k (a_2^1 - c_1^1)/(8 \cdot 4^{2n - 3}) - 2^{2n-1}
 - 8^{2 - 2n}/16 \},
\end{aligned} \\
c_1^1 + k (a_2^1 - c_1^1)/(8 \cdot 4^{2n - 3}) - 2^{2n-1}, \quad
k \in \{1, \dots, 8 \cdot 4^{2n - 3}\},\\
\begin{aligned} \min \{ &a_2^1 + (k-1)(b_2^1 - a_2^1)/(8 \cdot
4^{2n - 3}) - 2^{2n-1} + 8^{2 - 2n}/16, \\
&a_2^1 + k(b_2^1 - a_2^1)/(8 \cdot 4^{2n - 3}) - 2^{2n-1} \},
\end{aligned} \\
\begin{aligned}
\max \{ &a_2^1 + (k-1)(b_2^1 - a_2^1)/(8 \cdot 4^{2n - 3})
 - 2^{2n-1}, \\
&a_2^1 + k(b_2^1 - a_2^1)/(8 \cdot 4^{2n - 3}) - 2^{2n-1}
 - 8^{2 - 2n}/16 \},
\end{aligned} \\
 a_2^1 + k (b_2^1 - a_2^1)/(8 \cdot 4^{2n - 3}) - 2^{2n-1}, \quad
 k \in \{1, \dots, 8 \cdot 4^{2n - 3}\},\\
\begin{aligned}
 \min \{ &c_2^1 + (k - 1)(a_3^1 - c_2^1)/(8 \cdot 4^{2n - 3})
 - 2^{2n-1} + 8^{2 - 2n}/16, \\
&c_2^1 + k (a_3^1 - c_2^1)/(8 \cdot 4^{2n - 3}) - 2^{2n-1} \},
 \end{aligned} \\
\begin{aligned}
\max \{ &c_2^1 + (k - 1)(a_3^1 - c_2^1)/(8 \cdot 4^{2n - 3})
 - 2^{2n-1}, \\
&c_2^1 + k (a_3^1 - c_2^1)/(8 \cdot 4^{2n - 3}) - 2^{2n-1}
  - 8^{2 - 2n}/16 \},
\end{aligned} \\
 c_2^1 + k (a_3^1 - c_2^1)/(8 \cdot 4^{2n - 3}) - 2^{2n-1},
 \quad k \in \{1, \dots, 8 \cdot 4^{2n - 3}\},\\
 c_1^2 - 2^{2n-1}, \quad \min \{ c_1^2 - 2^{2n-1},
 b_1^2 - 2^{2n-1} + 8^{2 - 2n}/16 \},\\
\max \{ c_1^2 - 2^{2n-1} - 8^{2 - 2n}/16, b_1^2 - 2^{2n-1} \}, \\
\dots \\
c_6^2  - 2^{2n-1}, \quad \min \{ c_6^2 - 2^{2n-1},
  b_6^2 - 2^{2n-1} + 8^{2 - 2n}/16 \}, \\
 \max \{ c_6^2 - 2^{2n-1} - 8^{2 - 2n}/16, b_6^2 - 2^{2n-1} \}, \\
\begin{aligned}
 \min \{ &a_1^2 + (k-1)(b_1^2 - a_1^2)/(8 \cdot 4^{2n - 4})
 - 2^{2n-1} + 8^{2 - 2n}/16, \\
&a_1^2 + k(b_1^2 - a_1^2)/(8 \cdot 4^{2n - 4}) - 2^{2n-1} \},
\end{aligned}\\
\begin{aligned}
 \max \{ &a_1^2 + (k-1)(b_1^2 - a_1^2)/(8 \cdot 4^{2n - 4})
 - 2^{2n-1}, \\
&a_1^2 + k(b_1^2 - a_1^2)/(8 \cdot 4^{2n - 4}) - 2^{2n-1}
 - 8^{2 - 2n}/16 \},
\end{aligned}\\
a_1^2 + k (b_1^2 - a_1^2)/(8 \cdot 4^{2n - 4}) - 2^{2n-1},
 \quad k \in \{1, \dots, 8 \cdot 4^{2n - 4}\},\\
 \begin{aligned}
 \min \{ &c_1^2 + (k - 1)(a_2^2 - c_1^2)/(8 \cdot 4^{2n - 4})
 - 2^{2n-1} + 8^{2 - 2n}/16, \\
&c_1^2 + k (a_2^2 - c_1^2)/(8 \cdot 4^{2n - 4}) - 2^{2n-1} \},
\end{aligned} \\
\begin{aligned}
 \max \{ &c_1^2 + (k - 1)(a_2^2 - c_1^2)/(8 \cdot 4^{2n - 4})
  - 2^{2n-1}, \\
&c_1^2 + k (a_2^2 - c_1^2)/(8 \cdot 4^{2n - 4}) - 2^{2n-1}
 - 8^{2 - 2n}/16 \},
\end{aligned}\\
 c_1^2 + k (a_2^2 - c_1^2)/(8 \cdot 4^{2n - 4}) - 2^{2n-1},
 \quad k \in \{1, \dots, 8 \cdot 4^{2n - 4}\},\\
\dots \\
 \begin{aligned}
\min \{ &a_6^2 + (k-1)(b_6^2 - a_6^2)/(8 \cdot 4^{2n - 4})
 - 2^{2n-1} + 8^{2 - 2n}/16, \\
&a_6^2 + k(b_6^2 - a_6^2)/(8 \cdot 4^{2n - 4}) - 2^{2n-1} \},
 \end{aligned}\\
\begin{aligned}
 \max \{ &a_6^2 + (k-1)(b_6^2 - a_6^2)/(8 \cdot 4^{2n - 4})
 - 2^{2n-1}, \\
&a_6^2 + k(b_6^2 - a_6^2)/(8 \cdot 4^{2n - 4}) - 2^{2n-1}
 - 8^{2 - 2n}/16 \},
\end{aligned}\\
a_6^2 + k (b_6^2 - a_6^2)/(8 \cdot 4^{2n - 4}) - 2^{2n-1},
 \quad k \in \{1, \dots, 8 \cdot 4^{2n - 4}\},\\
\begin{aligned}
\min \{ &c_6^2 + (k - 1)(a_7^2 - c_6^2)/(8 \cdot 4^{2n - 4})
- 2^{2n-1} + 8^{2 - 2n}/16, \\
&c_6^2 + k (a_7^2 - c_6^2)/(8 \cdot 4^{2n - 4}) - 2^{2n-1} \},
 \end{aligned}\\
\begin{aligned}
 \max \{ &c_6^2 + (k - 1)(a_7^2 - c_6^2)/(8 \cdot 4^{2n - 4})
- 2^{2n-1}, \\
&c_6^2 + k (a_7^2 - c_6^2)/(8 \cdot 4^{2n - 4}) - 2^{2n-1}
- 8^{2 - 2n}/16 \},
\end{aligned}\\
c_6^2 + k (a_7^2 - c_6^2)/(8 \cdot 4^{2n - 4}) - 2^{2n-1}, \quad
 k \in \{1, \dots, 8 \cdot 4^{2n - 4}\}, \\
 \dots \\
 c_1^{2n-2} - 2^{2n-1}, \quad \min \{ c_1^{2n-2}
 - 2^{2n-1}, b_1^{2n-2} - 2^{2n-1} + 8^{2 - 2n}/16 \}, \\
 \max \{ c_1^{2n-2} - 2^{2n-1} - 8^{2 - 2n}/16, b_1^{2n-2}
 - 2^{2n-1} \}, \\
 \dots \\
 c_{7 \cdot 8^{2n -4}}^{2n-2}  - 2^{2n-1}, \quad
 \min \{ c_{7 \cdot 8^{2n -4}}^{2n-2} - 2^{2n-1},
 b_{7 \cdot 8^{2n -4}}^{2n-2} - 2^{2n-1} + 8^{2 - 2n} \}, \\
 \max \{ c_{7 \cdot 8^{2n -4}}^{2n-2} - 2^{2n-1}
 - 8^{2 - 2n}/16, b_{7 \cdot 8^{2n -4}}^{2n-2} - 2^{2n-1} \}, \\
\begin{aligned}
 \min \{ &a_1^{2n-2} + (k-1)(b_1^{2n-2} - a_1^{2n-2})/8 - 2^{2n-1}
+ 8^{2 - 2n}/16, \\
&a_1^{2n-2} + k(b_1^{2n-2} - a_1^{2n-2})/8 - 2^{2n-1} \},
\end{aligned}\\
\begin{aligned}
\max \{ &a_1^{2n-2} + (k-1)(b_1^{2n-2} - a_1^{2n-2})/8 - 2^{2n-1}, \\
&a_1^{2n-2} + k(b_1^{2n-2} - a_1^{2n-2})/8 - 2^{2n-1}
 - 8^{2 - 2n}/16 \},
\end{aligned}\\
a_1^{2n-2} + k (b_1^{2n-2} - a_1^{2n-2})/8 - 2^{2n-1}, \quad
k \in \{1, \dots, 8\},\\
\begin{aligned}
\min \{ &c_1^{2n-2} + (k - 1)(a_2^{2n-2} - c_1^{2n-2})/8 - 2^{2n-1}
+ 8^{2 - 2n}/16, \\
&c_1^{2n-2} + k (a_2^{2n-2} - c_1^{2n-2})/8 - 2^{2n-1} \},
\end{aligned} \\
\begin{aligned}
 \max \{ &c_1^{2n-2} + (k - 1)(a_2^{2n-2} - c_1^{2n-2})/8 - 2^{2n-1}, \\
&c_1^{2n-2} + k (a_2^{2n-2} - c_1^{2n-2})/8 - 2^{2n-1}
- 8^{2 - 2n}/16 \},
\end{aligned}\\
c_1^{2n-2} + k (a_2^{2n-2} - c_1^{2n-2})/8 - 2^{2n-1},
\quad k \in \{1, \dots, 8\},\\
\dots \\
\begin{aligned}
\min \{ &a_{7 \cdot 8^{2n -4}}^{2n-2}
+ (k-1)(b_{7 \cdot 8^{2n -4}}^{2n-2} - a_{7 \cdot 8^{2n -4}}^{2n-2})/8
- 2^{2n-1} + 8^{2 - 2n}/16, \\
&a_{7 \cdot 8^{2n -4}}^{2n-2} + k(b_{7 \cdot 8^{2n -4}}^{2n-2}
- a_{7 \cdot 8^{2n -4}}^{2n-2})/8 - 2^{2n-1} \},
\end{aligned}\\
\begin{aligned}
\max \{ &a_{7 \cdot 8^{2n -4}}^{2n-2} + (k-1)(b_{7 \cdot
8^{2n -4}}^{2n-2} - a_{7 \cdot 8^{2n -4}}^{2n-2})/8 - 2^{2n-1},  \\
&a_{7 \cdot 8^{2n -4}}^{2n-2} + k(b_{7 \cdot 8^{2n -4}}^{2n-2}
- a_{7 \cdot 8^{2n -4}}^{2n-2})/8 - 2^{2n-1} - 8^{2 - 2n}/16 \},
\end{aligned} \\
a_{7 \cdot 8^{2n -4}}^{2n-2} + k (b_{7 \cdot 8^{2n -4}}^{2n-2}
- a_{7 \cdot 8^{2n -4}}^{2n-2})/8 - 2^{2n-1}, \quad
k \in \{1, \dots, 8\},\\
\begin{aligned}
 \min \{ &c_{7 \cdot 8^{2n -4}}^{2n-2} + (k - 1)(a_{7 \cdot 8^{2n -4}
+ 1}^{2n-2} - c_{7 \cdot 8^{2n -4}}^{2n-2})/8 - 2^{2n-1}
+ 8^{2 - 2n}/16, \\
&c_{7 \cdot 8^{2n -4}}^{2n-2} + k (a_{7 \cdot 8^{2n -4} + 1}^{2n-2}
 - c_{7 \cdot 8^{2n -4}}^{2n-2})/8 - 2^{2n-1} \},
\end{aligned}\\
\begin{aligned}
\max \{ &c_{7 \cdot 8^{2n -4}}^{2n-2} + (k - 1)
 (a_{7 \cdot 8^{2n -4} + 1}^{2n-2} - c_{7 \cdot 8^{2n -4}}^{2n-2})/8
 - 2^{2n-1}, \\
&c_{7 \cdot 8^{2n -4}}^{2n-2} + k (a_{7 \cdot 8^{2n -4} + 1}^{2n-2}
 - c_{7 \cdot 8^{2n -4}}^{2n-2})/8 - 2^{2n-1}
 - 8^{2 - 2n}/16 \},
\end{aligned} \\
c_{7 \cdot 8^{2n -4}}^{2n-2} + k (a_{7 \cdot 8^{2n -4} + 1}^{2n-2}
- c_{7 \cdot 8^{2n -4}}^{2n-2})/8 - 2^{2n-1}, \quad
k \in \{1, \dots, 8\},
\end{gather*}
and from a number of $ b_1^1 - 2^{2n-1}$ such that the total
number of $d_l^{2n-1}$ is $21 \cdot 8^{2n -2}$. We denote
\begin{gather*}
 a_{1}^{2n} := d_1^{2n-1}, \quad b_1^{2n} := d_2^{2n-1}, \quad
c_1^{2n} := d_3^{2n-1}, \quad\dots \\
c_{7 \cdot 8^{2n -2}}^{2n-1} := d_{21 \cdot 8^{2n -2}}^3,
\quad a_{7 \cdot 8^{2n -2} + 1}^{2n-1} := -2^{2n - 3} - \dots -2.
\end{gather*}

In the $(2n + 1)$-th step, we define continuous $C$ so that
$$
\| C(t) - C(t - 2^{2n}) \| < \frac{\varepsilon}{2^{n+2}}, \quad
t\in (2 + \dots + 2^{2n - 2}, 2 + \dots + 2^{2n}].
$$
Now $C$ has constant values on
$[b^{2n+1}_j, c^{2n+1}_j]$, $j \in \{1, \dots, 7 \cdot 8^{2n - 1}\}$,
where we put
$$
a_{1}^{2n+1} := 2 + 2^2 + \dots + 2^{2n - 2}
$$
and we obtain
$$
b_1^{2n+1}, \quad c_1^{2n+1}, \quad a_2^{2n+1},
\quad \dots \quad c_{7 \cdot 8^{2n - 1}}^{2n+1},
\quad a_{7 \cdot 8^{2n - 1} + 1}^{2n+1}
$$
from the nondecreasing sequence of
\begin{gather*}
a_{j+1}^{2n} + 2^{2n}, \quad b_j^{2n} + 2^{2n}, \quad c_j^{2n}
 + 2^{2n}, \\
\min \{ a_j^{2n} + 2^{2n} + 8^{1 - 2n}/16, b_j^{2n} + 2^{2n} \}, \quad
 \max \{ a_j^{2n} + 2^{2n}, b_j^{2n} + 2^{2n} - 8^{1 - 2n}/16 \}, \\
 \min \{ c_j^{2n} + 2^{2n}, b_j^{2n} + 2^{2n} + 8^{1 - 2n}/16 \}, \quad
 \max \{ c_j^{2n} + 2^{2n} - 8^{1 - 2n}/16, b_j^{2n} + 2^{2n}\}, \\
 \min \{ c_j^{2n} + 2^{2n} + 8^{1 - 2n}/16, a_{j+1}^{2n} + 2^{2n}\},
 \quad \max \{ c_j^{2n} + 2^{2n}, a_{j+1}^{2n} + 2^{2n}
 - 8^{1 - 2n}/16 \}
\end{gather*}
for $j \in \{1, \dots, 7 \cdot 8^{2n -2}\}$ and
\begin{gather*}
 c_1^1  + 2^{2n}, \quad \min \{ c_1^1 + 2^{2n}, b_1^1 + 2^{2n}
 + 8^{1 - 2n}/16 \}, \\
 \max \{ c_1^1 + 2^{2n} - 8^{1 - 2n}/16, b_1^1 + 2^{2n}\}, \\
  c_2^1  + 2^{2n}, \quad \min \{ c_2^1 + 2^{2n}, b_2^1 + 2^{2n}
 + 8^{1 - 2n}/16 \}, \\
 \max \{ c_2^1 + 2^{2n} - 8^{1 - 2n}/16, b_2^1 + 2^{2n}\}, \\
\begin{aligned}
\min \{ &a_1^1 + (k-1)(b_1^1 - a_1^1)/(8 \cdot 4^{2n - 2}) + 2^{2n}
 + 8^{1 - 2n}/16, \\
&a_1^1 + k(b_1^1 - a_1^1)/(8 \cdot 4^{2n - 2}) + 2^{2n} \},
\end{aligned}
\\
\begin{aligned}
\max \{ &a_1^1 + (k-1)(b_1^1 - a_1^1)/(8 \cdot 4^{2n - 2}) + 2^{2n}, \\
&a_1^1 + k(b_1^1 - a_1^1)/(8 \cdot 4^{2n - 2}) + 2^{2n}
 - 8^{1 - 2n}/16 \},
\end{aligned} \\
 a_1^1 + k (b_1^1 - a_1^1)/(8 \cdot 4^{2n - 2}) + 2^{2n}, \quad
k \in \{1, \dots, 8 \cdot 4^{2n - 2}\},\\
\begin{aligned}
\min \{ &c_1^1 + (k - 1)(a_2^1 - c_1^1)/(8 \cdot 4^{2n - 2})
 + 2^{2n} + 8^{1 - 2n}/16, \\
&c_1^1 + k (a_2^1 - c_1^1)/(8 \cdot 4^{2n - 2}) + 2^{2n} \},
\end{aligned}
\\
\begin{aligned}
\max \{ &c_1^1 + (k - 1)(a_2^1 - c_1^1)/(8 \cdot 4^{2n - 2})
 + 2^{2n}, \\
&c_1^1 + k (a_2^1 - c_1^1)/(8 \cdot 4^{2n - 2}) + 2^{2n}
 - 8^{1 - 2n}/16 \},
\end{aligned} \\
c_1^1 + k (a_2^1 - c_1^1)/(8 \cdot 4^{2n - 2}) + 2^{2n}, \quad
 k \in \{1, \dots, 8 \cdot 4^{2n - 2}\},\\
\begin{aligned}
\min \{ &a_2^1 + (k-1)(b_2^1 - a_2^1)/(8 \cdot 4^{2n - 2})
+ 2^{2n} + 8^{1 - 2n}/16, \\
&a_2^1 + k(b_2^1 - a_2^1)/(8 \cdot 4^{2n - 2}) + 2^{2n} \},
\end{aligned}
\\
\begin{aligned}
\max \{ &a_2^1 + (k-1)(b_2^1 - a_2^1)/(8 \cdot 4^{2n - 2})
+ 2^{2n}, \\ &a_2^1 + k(b_2^1 - a_2^1)/(8 \cdot 4^{2n - 2})
+ 2^{2n} - 8^{1 - 2n}/16 \},
\end{aligned}\\
 a_2^1 + k (b_2^1 - a_2^1)/(8 \cdot 4^{2n - 2}) + 2^{2n}, \quad
k \in \{1, \dots, 8 \cdot 4^{2n - 2}\},\\
\begin{aligned}
 \min \{ &c_2^1 + (k - 1)(a_3^1 - c_2^1)/(8 \cdot 4^{2n - 2})
+ 2^{2n} + 8^{1 - 2n}/16, \\
&c_2^1 + k (a_3^1 - c_2^1)/(8 \cdot 4^{2n - 2}) + 2^{2n} \},
\end{aligned}
\\
\begin{aligned}
 \max \{ &c_2^1 + (k - 1)(a_3^1 - c_2^1)/(8 \cdot 4^{2n - 2})
+ 2^{2n}, \\ &c_2^1 + k (a_3^1 - c_2^1)/(8 \cdot 4^{2n - 2})
+ 2^{2n} - 8^{1 - 2n}/16 \},
 \end{aligned}\\
 c_2^1 + k (a_3^1 - c_2^1)/(8 \cdot 4^{2n - 2}) + 2^{2n}, \quad
k \in \{1, \dots, 8 \cdot 4^{2n - 2}\},\\
 c_1^2 + 2^{2n}, \quad \min \{ c_1^2 + 2^{2n}, b_1^2 + 2^{2n}
+ 8^{1 - 2n}/16 \}, \\
 \max \{ c_1^2 + 2^{2n} - 8^{1 - 2n}/16, b_1^2 + 2^{2n} \},\\
\dots \\
c_6^2  + 2^{2n}, \quad \min \{ c_6^2 + 2^{2n}, b_6^2 + 2^{2n}
 + 8^{1 - 2n}/16 \}, \\
 \max \{ c_6^2 + 2^{2n} - 8^{1 - 2n}/16, b_6^2 + 2^{2n} \},
 \\
\begin{aligned}
\min \{ &a_1^2 + (k-1)(b_1^2 - a_1^2)/(8 \cdot 4^{2n - 3})
+ 2^{2n} + 8^{1 - 2n}/16, \\
&a_1^2 + k(b_1^2 - a_1^2)/(8 \cdot 4^{2n - 3}) + 2^{2n} \},
\end{aligned} \\
\begin{aligned}
\max \{ &a_1^2 + (k-1)(b_1^2 - a_1^2)/(8 \cdot 4^{2n - 3})
+ 2^{2n}, \\
&a_1^2 + k(b_1^2 - a_1^2)/(8 \cdot 4^{2n - 3}) + 2^{2n}
- 8^{1 - 2n}/16 \},
\end{aligned}\\
a_1^2 + k (b_1^2 - a_1^2)/(8 \cdot 4^{2n - 3}) + 2^{2n},
 \quad k \in \{1, \dots, 8 \cdot 4^{2n - 3}\},\\
\begin{aligned}
\min \{ &c_1^2 + (k - 1)(a_2^2 - c_1^2)/(8 \cdot 4^{2n - 3})
+ 2^{2n} + 8^{1 - 2n}/16, \\ &c_1^2
 + k (a_2^2 - c_1^2)/(8 \cdot 4^{2n - 3})
 + 2^{2n} \}, \end{aligned}
\\
\begin{aligned}
\max \{ &c_1^2 + (k - 1)(a_2^2 - c_1^2)/(8 \cdot 4^{2n - 3})
+ 2^{2n}, \\
&c_1^2 + k (a_2^2 - c_1^2)/(8 \cdot 4^{2n - 3}) + 2^{2n}
 - 8^{1 - 2n}/16 \},
\end{aligned}\\
c_1^2 + k (a_2^2 - c_1^2)/(8 \cdot 4^{2n - 3}) + 2^{2n}, \quad
k \in \{1, \dots, 8 \cdot 4^{2n - 3}\},\\
 \dots \\
\begin{aligned}
\min \{ &a_6^2 + (k-1)(b_6^2 - a_6^2)/(8 \cdot 4^{2n - 3})
+ 2^{2n} + 8^{1 - 2n}/16, \\ &a_6^2 + k(b_6^2
- a_6^2)/(8 \cdot 4^{2n - 3}) + 2^{2n} \},
\end{aligned} \\
\begin{aligned}
\max \{ &a_6^2 + (k-1)(b_6^2 - a_6^2)/(8 \cdot 4^{2n - 3})
+ 2^{2n}, \\ &a_6^2 + k(b_6^2 - a_6^2)/(8 \cdot 4^{2n - 3})
+ 2^{2n} - 8^{1 - 2n}/16 \},
\end{aligned}\\
a_6^2 + k (b_6^2 - a_6^2)/(8 \cdot 4^{2n - 3}) + 2^{2n}, \quad
k \in \{1, \dots, 8 \cdot 4^{2n - 3}\},\\
\begin{aligned}
 \min \{ &c_6^2 + (k - 1)(a_7^2 - c_6^2)/(8 \cdot 4^{2n - 3})
+ 2^{2n} + 8^{1 - 2n}/16, \\ &c_6^2 + k (a_7^2
- c_6^2)/(8 \cdot 4^{2n - 3}) + 2^{2n} \},
\end{aligned}
\\
\begin{aligned}
\max \{ &c_6^2 + (k - 1)(a_7^2 - c_6^2)/(8 \cdot 4^{2n - 3})
+ 2^{2n}, \\
&c_6^2 + k (a_7^2 - c_6^2)/(8 \cdot 4^{2n - 3}) + 2^{2n}
- 8^{1 - 2n}/16 \},
\end{aligned}
\\
 c_6^2 + k (a_7^2 - c_6^2)/(8 \cdot 4^{2n - 3}) + 2^{2n}, \quad
k \in \{1, \dots, 8 \cdot 4^{2n - 3}\}, \\
 \dots \\
 c_1^{2n - 1} + 2^{2n}, \quad \min \{ c_1^{2n - 1} + 2^{2n},
 b_1^{2n - 1} + 2^{2n} + 8^{1 - 2n}/16 \}, \\
\max \{ c_1^{2n - 1} + 2^{2n} - 8^{1 - 2n}/16, b_1^{2n - 1}
+ 2^{2n} \}, \\
\dots \\
c_{7 \cdot 8^{2n - 3}}^{2n - 1}  + 2^{2n}, \quad
 \min \{ c_{7 \cdot 8^{2n - 3}}^{2n - 1} + 2^{2n},
 b_{7 \cdot 8^{2n - 3}}^{2n - 1} + 2^{2n} + 8^{1 - 2n}/16 \}, \\
 \max \{ c_{7 \cdot 8^{2n -3}}^{2n - 1} + 2^{2n} - 8^{1 - 2n}/16,
 b_{7 \cdot 8^{2n -3}}^{2n - 1} + 2^{2n} \}, \\
\begin{aligned}
\min \{ &a_1^{2n - 1} + (k-1)(b_1^{2n - 1}
- a_1^{2n - 1})/8 + 2^{2n} + 8^{1 - 2n}/16, \\ &a_1^{2n - 1}
+ k(b_1^{2n - 1} - a_1^{2n - 1})/8 + 2^{2n} \},
\end{aligned} \\
\begin{aligned}
 \max \{ &a_1^{2n - 1} + (k-1)(b_1^{2n - 1} - a_1^{2n - 1})/8 + 2^{2n},
 \\
&a_1^{2n - 1} + k(b_1^{2n - 1} - a_1^{2n - 1})/8 + 2^{2n}
- 8^{1 - 2n}/16 \},
\end{aligned}\\
a_1^{2n - 1} + k (b_1^{2n - 1} - a_1^{2n - 1})/8 + 2^{2n}, \quad
k \in \{1, \dots, 8\},\\
\begin{aligned}
\min \{ &c_1^{2n - 1} + (k - 1)(a_2^{2n - 1} - c_1^{2n - 1})/8
+ 2^{2n} + 8^{1 - 2n}/16, \\
 &c_1^{2n - 1} + k (a_2^{2n - 1} - c_1^{2n - 1})/8 + 2^{2n} \},
 \end{aligned} \\
\begin{aligned}
\max \{ &c_1^{2n - 1} + (k - 1)(a_2^{2n - 1} - c_1^{2n - 1})/8
  + 2^{2n},\\ &c_1^{2n - 1} + k (a_2^{2n - 1} - c_1^{2n - 1})/8
+ 2^{2n} - 8^{1 - 2n}/16 \},
\end{aligned}\\
 c_1^{2n - 1} + k (a_2^{2n - 1} - c_1^{2n - 1})/8 + 2^{2n}, \quad
 k \in \{1, \dots, 8\},\\
 \dots \\
\begin{aligned}
\min \{ &a_{7 \cdot 8^{2n - 3}}^{2n - 1}
+ (k-1)(b_{7 \cdot 8^{2n - 3}}^{2n - 1}
- a_{7 \cdot 8^{2n - 3}}^{2n - 1})/8
+ 2^{2n} + 8^{1 - 2n}/16, \\
&a_{7 \cdot 8^{2n - 3}}^{2n - 1} + k(b_{7 \cdot 8^{2n - 3}}^{2n - 1}
- a_{7 \cdot 8^{2n - 3}}^{2n - 1})/8 + 2^{2n} \},
\end{aligned}\\
\begin{aligned}
\max \{ &a_{7 \cdot 8^{2n - 3}}^{2n - 1}
+ (k-1)(b_{7 \cdot 8^{2n - 3}}^{2n - 1} - a_{7 \cdot
8^{2n - 3}}^{2n - 1})/8 + 2^{2n}, \\
&a_{7 \cdot 8^{2n - 3}}^{2n - 1} + k(b_{7 \cdot 8^{2n - 3}}^{2n - 1}
- a_{7 \cdot 8^{2n - 3}}^{2n - 1})/8 + 2^{2n}
- 8^{1 - 2n}/16 \},
\end{aligned} \\
a_{7 \cdot 8^{2n - 3}}^{2n - 1} + k (b_{7 \cdot 8^{2n - 3}}^{2n - 1}
 - a_{7 \cdot 8^{2n - 3}}^{2n - 1})/8 + 2^{2n}, \quad
k \in \{1, \dots, 8\},\\
\begin{aligned}
\min \{ &c_{7 \cdot 8^{2n - 3}}^{2n - 1} + (k - 1)(a_{7 \cdot
 8^{2n - 3} + 1}^{2n - 1} - c_{7 \cdot 8^{2n - 3}}^{2n - 1})/8
 + 2^{2n} + 8^{1 - 2n}/16, \\
&c_{7 \cdot 8^{2n - 3}}^{2n - 1}
 + k (a_{7 \cdot 8^{2n - 3} + 1}^{2n - 1} - c_{7 \cdot
8^{2n - 3}}^{2n - 1})/8 + 2^{2n} \},
\end{aligned} \\
\begin{aligned} \max \{ &c_{7 \cdot 8^{2n - 3}}^{2n - 1}
+ (k - 1)(a_{7 \cdot 8^{2n - 3} + 1}^{2n - 1}
- c_{7 \cdot 8^{2n - 3}}^{2n - 1})/8 + 2^{2n}, \\
&c_{7 \cdot 8^{2n - 3}}^{2n - 1} + k (a_{7 \cdot 8^{2n - 3}
+ 1}^{2n - 1} - c_{7 \cdot 8^{2n - 3}}^{2n - 1})/8 + 2^{2n}
- 8^{1 - 2n}/16 \},
\end{aligned} \\
c_{7 \cdot 8^{2n - 3}}^{2n - 1} + k (a_{7 \cdot 8^{2n - 3}
+ 1}^{2n - 1} - c_{7 \cdot 8^{2n - 3}}^{2n - 1})/8 + 2^{2n}, \quad
k \in \{1, \dots, 8\},
\end{gather*}
and the corresponding number of $ b_1^1 + 2^{2n}$.

Using this construction, we obtain a continuous function $C$ on
$\mathbb{R}$.
 From Theorem \ref{th3.v} it follows that $C$ is almost periodic.
Since
\begin{gather*}
 \| C(t) \| = 0, \quad t \in [0, 1], \quad
\| C(t) \| < \varepsilon/4, \quad t \in (1, 2],\\
\| C(t) - C(t + 2) \| < \varepsilon/4, \quad t \in [-2, 0),
\quad \| C(t) - C(t - 4)\| < \varepsilon/8,
\quad t \in (2, 6], \\
\dots \\
\begin{aligned}
&\| C(t) - C(t + 2^{2n-1})\| < \varepsilon/2^{n+1}, \\
&t \in [-2^{2n - 1} - \dots - 2^3 -2, -2^{2n - 3} - \dots - 2^3 -2),
\end{aligned} \\
\| C(t) - C(t - 2^{2n}) \| < \varepsilon/2^{n+2}, \quad t \in
(2 + 2^2 + \dots + 2^{2n - 2}, 2 + 2^2 + \dots + 2^{2n}],
\end{gather*}
we see that
\begin{equation} \label{ves0ht4}
\|C(t) \| < \sum_{j=1}^\infty
\frac{2 \varepsilon}{2^{j + 1}} = \varepsilon, \quad t \in \mathbb{R}.
\end{equation}

We denote
$$
I_n := [2 + 2^2 + \dots + 2^{2n - 2}, 2 + 2^2 + \dots + 2^{2n}].
$$
We will prove that we can choose constant values of $C (t)$,
$t \in I_n$ on subintervals with
the total length denoted by $r_{2n + 1}$ which is grater than
$2^{2n - 1}$ for all $n \in \mathbb{N}$. We can choose $C$ on
\begin{gather*}
[4 -2 + 1/16 + 8^{-1}/16, 4 -2 + 1 - 1/16 - 8^{-1}/16]
\subset [2, 6],\\
[4 -2 + 1 + 1/4 + 1/16 + 8^{-1}/16, 4 -2 + 1 + 3/4 - 1/16
 - 8^{-1}/16] \subset [2, 6],\\
[4 + 8^{-1}/16, 4 + 1 - 8^{-1}/16],
 [4 + 1 + 1/4 + 8^{-1}/16, 4 + 1 + 3/4 - 8^{-1}/16] \subset [2, 6].
\end{gather*}
Hence,
\begin{equation} \label{vesoii368}
r_3 \ge 55/64 + 23/64 + 63/64 + 31/64 = 43/16;
\end{equation}
i.e., the statement is valid for $n = 1$.
We use the induction principle with respect to~$n$.
Assume that the
statement is true for $1, 2, \dots, n - 1$ and prove it for $n$.
Without loss of the generality (consider the below given process),
we can also assume that
the estimation $r_{2j} > 2^{2(j - 1)}$ is valid
for $j \in \{1, \dots, n\}$ (note that $r_2 = 5/4 > 2^0$)
if we use analogous notation.

In view of the construction, we see that we can choose $C$
on any interval
$$
[s + 2^{2n} + 8^{1 - 2n}/16, t + 2^{2n} - 8^{1 - 2n}/16]
$$
if we can choose $C$ on $[s, t]$, where $s = b_j^{l} < c_j^{l} = t$,
$l < 2n + 1$. Especially, we can choose $C$ on
\begin{gather*}
[2^{2n} + 8^{1 - 2n}/16, 1 + 2^{2n} - 8^{1 - 2n}/16],\\
[1 + 1/4 + 2^{2n} + 8^{1 - 2n}/16, 1 + 3/4 + 2^{2n} - 8^{1 - 2n}/16],\\
[-2 + 1/16 + 2^{2n} + 8^{1 - 2n}/16, -2 + 1 - 1/16 + 2^{2n}
- 8^{1 - 2n}/16],\\
[-2 + 1 + 1/4 + 1/16 + 2^{2n} + 8^{1 - 2n}/16, -2 + 1 + 3/4 -1/16
 + 2^{2n} - 8^{1 - 2n}/16]
\end{gather*}
and on less than $7 \cdot 8^{2n -1} - 4$ subintervals of $I_n$.
Expressing
\begin{equation} \label{ves041}
\begin{aligned}
I_n = &[0 + 2^{2n}, 1 + 2^{2n}] \cup [1 + 2^{2n}, 2 + 2^{2n}]
\cup [-2 + 2^{2n}, 0 + 2^{2n}] \cup \dots \\
&\cup [2 + 2^2 + \dots + 2^{2n - 4} + 2^{2n}, 2 + 2^2 + \dots
 + 2^{2n -2} + 2^{2n}]\\
&\cup [-2^{2n - 1} - \dots - 2^3 -2 + 2^{2n}, -2^{2n - 3}
- \dots - 2^3 -2 + 2^{2n}]
\end{aligned}
\end{equation}
and using the induction hypothesis, the construction,
and \eqref{vesoii368}, we obtain that we can choose $C$
on intervals of the lengths grater than or equal to
\begin{gather*}
 1 - 2 \cdot 8^{1 - 2n}/16, \quad 1/2 - 2 \cdot 8^{1 - 2n}/16,\\
1 - 1/8 - 2 \cdot 8^{1 - 2n}/16,\quad  1/2 - 1/8 - 2
 \cdot 8^{1 - 2n}/16, \\
 43/16 + 2^2 + 2^3 + \dots + 2^{2n - 3} + 2^{2n - 2}
- 2 \cdot 8^{1 - 2n}/16 \cdot (7 \cdot 8^{2n -1} - 4).
\end{gather*}
Summing, we obtain
\begin{equation} \label{ves042}
 r_{2n + 1} \ge 1 + \frac{1}{2} + \frac{7}{8} + \frac{3}{8} +
\frac{11}{16} + 2^{2n - 1} - 2 - \frac{7}{8} > 2^{2n - 1},
\end{equation}
which is the above statement. Analogously, we can prove
\begin{equation} \label{ves033o}
 r_{2n} > 2^{2n - 2}, \quad n \in \mathbb{N}.
\end{equation}

Now we describe the principal fundamental matrix $X_C$ on
$I_n$ for arbitra\-ry~$n \in \mathbb{N}$.
Since $C$ is constant a has the form
$ \operatorname{diag}[ia, ia, \dots, ia ]$
for some $a \in \mathbb{R}$
on each interval $[b^{2n+1}_j, c^{2n+1}_j]$,
$j \in \{1, \dots, 6 \cdot 4^{2n -1}\}$,
from
$$
X_C (t_2) - X_C (t_1) = \int_{t_1}^{t_2} C(\tau) X_C (\tau)\,d\tau,
\quad t_1, t_2 \in \mathbb{R},
$$
we obtain
\begin{equation}  \label{ves0yt6}
\begin{aligned}
&\| X_C (t) - {X}_C^{2n+1} (t) \| \le  \\
&\sum_{j=1}^{k}
\Big(\int_{a^{2n+1}_j}^{b^{2n+1}_j} \| C(\tau) X_C (\tau) \|d\tau
+ \int_{c^{2n+1}_j}^{a^{2n+1}_{j+1}}
\| C(\tau) X_C (\tau) \|d\tau\Big)
\end{aligned}
\end{equation}
if $t \le a^{2n+1}_{k+1}$, $t\in I_n$, where
\begin{gather*}
 {X}_C^{2n+1} (t) := {X_C} (2 + 2^2 + \dots + 2^{2n - 2}), \quad
t \in [2 + 2^2 + \dots + 2^{2n - 2}, b^{2n + 1}_1],\\
 {X}_C^{2n+1} (t) := \exp (C(b^{2n+1}_1) (t - b^{2n+1}_1))
 \cdot {X_C^{2n+1}}
(b^{2n + 1}_1), \,\, t \in (b^{2n+1}_1, c^{2n+1}_1], \\
 {X}_C^{2n+1} (t) := {X_C}^{2n+1} (c^{2n+1}_1), \quad
t \in (c^{2n+1}_1, b^{2n + 1}_2],\\
 \dots \\
\begin{aligned}
{X}_C^{2n+1} (t) &:= \exp (C(b^{2n+1}_{7 \cdot 8^{2n -1}})
(t - b^{2n+1}_{7 \cdot 8^{2n -1}})) \cdot {X_C^{2n+1}}
(b^{2n+1}_{7 \cdot 8^{2n -1}}), \\
&t \in (b^{2n+1}_{7 \cdot 8^{2n -1}}, c^{2n+1}_{7 \cdot 8^{2n -1}}],
\end{aligned} \\
{X}_C^{2n+1} (t) := {X_C^{2n+1}} (c^{2n+1}_{7 \cdot 8^{2n -1}}), \quad
t \in (c^{2n+1}_{7 \cdot 8^{2n -1}}, 2 + 2^2 + \dots + 2^{2n}].
\end{gather*}
It is seen that $X_C$ is bounded (see also the below given,
where it is shown that
$X_C (t) \in U(m)$ for all~$t$) as almost periodic $C$.
Any interval
$$
[2 + \dots + 2^{2n - 2} + l - 1, 2 + \dots + 2^{2n - 2} + l ], \quad
l \in \{1, \dots, 2^{2n}\},\, n \in \mathbb{N}
$$
contains at most $4^{2n + 1}$ subintervals where $C$ can be linear.
Indeed, it suffices to consider the construction.
We repeat that the length of each one of the considered subintervals
is $8^{1 - 2n}/16$ which implies that the total length of them on
$$
J_n^l := [2 + 2^2 + \dots + 2^{2n - 2}, 2 + 2^2 + \dots
+ 2^{2n - 2} + 2^{2n - l}], \quad l \in \{1, \dots, n\}
$$
is less than~$2^{1-l}$. Thus (consider also \eqref{ves0yt6}),
there exists $K \in \mathbb{R}$
such that
\begin{equation} \label{ves02j9}
 \| X_C (t) - {X}_C^{2n+1} (t) \| \le \frac{K}{2^l},
\quad t \in J_n^l, \, l \in \{1, \dots, n\}, \, n \in \mathbb{N}.
\end{equation}
 From the form
$ \operatorname{diag} [ia(t), \dots, ia(t)]$
of all matrices $C(t)$, we see that
$$
\| C (t) \| = |a(t)|, \quad t\in \mathbb{R}.
 $$
For simplicity, let $a(t) \ge 0$, $t \in \mathbb{R}$.
Let $ a_j^n \in \mathbb{R}$, $j \in \{1, \dots, n\}$ be
arbitrarily chosen.
Considering the construction and combining \eqref{ves042}
and \eqref{ves033o},
we obtain that we can choose constant values of $C (t)$,
$t \in [2 + \dots + 2^{2n - 2} + (l - 1) 2^{n}, 2 + \dots
+ 2^{2n - 2} + l\, 2^{n} ]$ on subintervals with
the total length grater than $2^{n - 2}$
for each $l \in \{1, \dots, 2^n\}$ and all sufficiently
large $n \in \mathbb{N}$. Since we choose $C$
only so that
$$
\| C(t) - C(t - 2^{2n})\| < \varepsilon/2^{n+2}, \quad t \in I_n,
$$
we see that we can obtain
$$
{X}_C^{2n+1} (t_j^n) = \operatorname{diag}[\exp(ia_j^n), \dots, \exp(ia_j^n)]
$$
for arbitrary $t_j^n$ such that
\begin{equation}  \label{vesj52d} \begin{gathered}
 t_1^n \ge 2 + 2^2 + 2^4 + \dots + 2^{2n - 2} + 3^{n} - 3^0, \quad
 t_2^n \ge t_1^n + 3^{n} - 3^1, \\
 \dots \quad t_n^n \ge t_{n-1}^n + 3^{n} - 3^{n-1}, \quad
2 + 2^2 + 2^4 + \dots + 2^{2n} \ge t_n^n
\end{gathered}
\end{equation}
because we have
$$
4^n > n(3^{n} - 3^{0}) > 3^{n} - 3^{0} > \dots
> 3^{n} - 3^{n-1} > 2^{2n - k + 1}
$$
for sufficiently large $n \in \mathbb{N}$
and some $k = k (n) \in \{1, \dots, n\}$ satisfying
$$
2^{2n - k - 2} \cdot \varepsilon \cdot 2^{-n - 2}  > 2 \pi.
$$
We recall that we need to prove the existence of such $C$,
given by the above construction,
for which the vector valued function
$X_A (t) \cdot X_C (t) \cdot u$, $t \in \mathbb{R}$ is not
almost periodic for any $ u \in \mathbb{C}^m$, $\|u\|_1 = 1$.
Since
$$
(X_A (t)  X_A^* (t) )' = A(t)\, X_A (t) \, X_A^* (t)
- X_A (t) \, X_A^* (t) \, A(t), \quad t\in \mathbb{R}
$$
and since the constant function given by $I$ is a solution of
$X' = A \cdot X - X\cdot A$, $X(0)= I$, we have $X_A (t) \in U(m)$
for all~$t$.
Thus, $X_C (t) \in U(m)$, $t\in \mathbb{R}$ as well. We add that
$X_A (t) \cdot X_A^* (t) = I$, $t \in \mathbb{R}$ implies
$A^* (t) + A (t) = 0$, $t \in \mathbb{R}$.

Let $c\in \mathbb{C}$, $|c|=1$, and $N \in U(m)$ be arbitrarily
given. Obviously, for any $M \in U(m)$, we can choose a number $a
({M, c}) \in [0, 2\pi)$ in order that all eigenvalues of matrix
$P:= M \cdot \operatorname{diag} [\exp(ia({M, c})), \dots,
\exp(ia({M, c})) ]$ are not in the neighbourhood of $c$ with
a given radius which depends only on dimension $m$. Indeed, if $M$
has eigenvalues $\lambda_1, \dots, \lambda_{m}$, then the
eigenvalues of $P$ are $\lambda_1 \exp(ia({M, c}))$, $\dots$,
$\lambda_m \exp(ia({M, c}))$. Considering
$$
M  \operatorname{diag} [\exp(ia({M, c})), \dots, \exp(ia({M, c})) ]
 u - N  u
$$
and expressing vectors $u \in \mathbb{C}^m$, $|u_1| + \dots + |u_m| = 1$, 
as linear combinations of the eigenvectors of $P$,
we see that
$$
M \operatorname{diag} [\exp(ia({M, c})), \dots, \exp(ia({M, c}))] u
$$
cannot be in a neighbourhood of $N \cdot u$ for
some $c\in \mathbb{C}$, $|c|=1$. Thus (the considered
multiplication of matrices and vectors is uniformly continuous),
there exist $\vartheta>0$ and $\xi > 0$ such that, for any
matrices $M, N \in U(m)$, one can find $a ({M, N}) \in [0, 2\pi)$
satisfying
\begin{equation}  \label{ves01a6}
\begin{gathered}
 \| M  \operatorname{diag} [\exp(i\tilde{a}), \dots,
\exp(i\tilde{a}) ]  u  - N  u \|_1 > \vartheta, \\
  u \in \mathbb{C}^m, \| u\|_1 = 1, \quad
\tilde{a} \in (a({M, N}) - \xi,a({M, N}) + \xi).
\end{gathered}
\end{equation}
We showed that we can construct $C$ so that we obtain
$$
{X}_C^{2n+1} (t_j^n) =
\operatorname{diag} [\exp(ia_j^n), \dots, \exp(ia_j^n) ]
$$
for arbitrarily given $a_j^n \in [0, 2\pi)$ and any $t_j^n$
satisfying \eqref{vesj52d} if $n \in \mathbb{N}$ is sufficiently
large and $j \in \{1, \dots, n\}$.
Especially, for sufficiently large $n \in \mathbb{N}$ and for
\begin{equation}  \label{ves77l3}
\begin{gathered}
t_1^n := 2 + 2^2 + 2^4 + \dots + 2^{2n - 2} + 3^{n} - 3^{0},\\
t_2^n := t_1^n + 3^{n} - 3^{1}, \quad \dots \quad
t_n^n := t_{n-1}^n + 3^{n} - 3^{n - 1},
\end{gathered}
\end{equation}
we can choose all ${X}_C^{2n+1} (t_j^n)$ in the form
without any conditions. Hence, we obtain diagonal matrices
${X}_C^{2n+1} (t_j^n)$, $j \in \{1, \dots, n\}$, given by
numbers
$$
\exp \Big(i a (X_A (t_j^n), X_A (t_j^n - 3^n + 3^{j-1})
 X_C (t_j^n - 3^n + 3^{j-1}))\Big)
$$
on their diagonals.

It is seen from \eqref{ves77l3} that each
$$
t_j^n \in [2 + 2^2 + \dots + 2^{2n - 2}, 2 + 2^2
+ \dots + 2^{2n - 2} + n 3^{n}].
$$
Thus (see \eqref{ves02j9}), for any $\eta >0$, we have
\begin{equation} \label{ves0155}
 \| X_C (t_j^n) - {X}_C^{2n+1} (t_j^n) \| < \eta
\end{equation}
for sufficiently large $n = n (\eta) \in \mathbb{N}$ and
$j \in \{1, \dots, n\}$.
 From \eqref{ves01a6} and \eqref{ves0155} it follows that
\begin{equation} \label{ves01bq}
\| X_A (t_j^n) \,
X_C (t_j^n)  u  - X_A (t_j^n - 3^n + 3^{j-1})
 X_C (t_j^n - 3^n + 3^{j-1})  u \|_1 > \vartheta
\end{equation}
for $u \in \mathbb{C}^m$, $\| u\|_1 = 1$, sufficiently
large $n \in \mathbb{N}$, and $j \in \{1, \dots, n\}$.

By contradiction, suppose that there exists $ u \in \mathbb{C}^m$,
$\|u\|_1 = 1$, with the property that $X_A (t) \cdot X_C (t) \cdot u$,
$t\in \mathbb{R}$ is almost periodic.
Applying Theorem \ref{thm21} for
$$
\psi (t) = X_A (t)  X_C (t)  u, \quad t\in \mathbb{R},\;
s_n  = 3^n, \; n \in \mathbb{N}, \; \varepsilon = \vartheta,$$
we obtain
\begin{equation} \label{vesjtbq}
\| X_A (t + 3^{n_1})
X_C (t + 3^{n_1})  u  - X_A (t + 3^{n_2})  X_C (t + 3^{n_2})  u \|_1
< \vartheta, \quad t \in \mathbb{R}
\end{equation}
for all $n_1, n_2$ from an infinite set
$N(\vartheta) \subseteq \mathbb{N}$. If we rewrite \eqref{vesjtbq}
as
$$
\| X_A (t) X_C (t)  u  - X_A (t + 3^{n_2} - 3^{n_1})  X_C (t + 3^{n_2}
- 3^{n_1})  u \|_1 < \vartheta, \quad t \in \mathbb{R},
 $$
then it is easy to see that \eqref{ves01bq} is not valid
for infinitely many $n \in \mathbb{N}$.
This contradiction proves the theorem.
 \end{proof}

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\end{document}