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A subset $A \subset \mathbb{R}$ is relative dense if there exists a real number $L>0$ such that for every $t \in \mathbb{R}$ the set $A \cap [t,t+L]$ is not empty. Such number $L$ is called an inclusion interval.

A real number $\omega_\varepsilon \in \mathbb{R}$ is an $\varepsilon$-almost period for a function $f$ if $\sup\limits_{t \in \mathbb{R}}|f(t+\omega_{\varepsilon})-f(t)|<\varepsilon.$ Denote by $\Omega_{\varepsilon}(f)$ the set of all $\varepsilon$-almost periods of function $f$.

A continuous function $f: \mathbb{R} \to \mathbb{C}$ is almost periodic if for every $\varepsilon>0$ the set $\Omega_{\varepsilon}(f)$ is relative dense.

For a given almost periodic function $f$ denote by $L_{+}(\varepsilon)$ the infinum of the set of inclusion intervals for $\Omega_{\varepsilon}(f)$, i.e. $$L_{+}(\varepsilon) := \inf\limits\{L_{\varepsilon} > 0 \ : \ L_{\varepsilon} \text{can be chosen as inclusion interval for } \Omega_{\varepsilon}(f) \}.$$

Now consider an almost periodic function $f$. The set $\Omega_{\varepsilon}(f)$ is symmetric, i.e. if $\omega_{\varepsilon} \in \Omega_{\varepsilon}(f)$ then $-\omega_{\varepsilon} \in \Omega_{\varepsilon}(f)$. Also $0 \in \Omega_{\varepsilon}(f)$ and due to uniform continuity there exists the maximal interval $\Delta_{0}(\varepsilon) \ni 0$ such that $\Delta_0(\varepsilon) \subset \Omega_{\varepsilon}(f)$. For sufficiently small $\varepsilon$ the interval $\Delta_{0}(\varepsilon)$ is finite length and it is separated from the others almost periods.

I'm interested in bounds of the distance between $\Delta_0(\varepsilon)$ and the first apperance of almost periods outside of $\Delta_0(\varepsilon)$ in the terms of $L_{+}(\varepsilon)$.

We say that an almost periodic function $f$ satisfies the property A if there exists numbers $\varepsilon_0 > 0, \ C_1>0, \ C_2>0$ such that for every $\varepsilon \in (0;\varepsilon_0)$ $[-\frac{L_{+}(C_1\varepsilon)}{C_2};\frac{L_{+}(C_1\varepsilon)}{C_2}] \cap \Omega_{\varepsilon}(f) = \Delta_0(\varepsilon)$, i.e. theres no $\varepsilon$-almost periods in the interval $[-\frac{L_{+}(C_1\varepsilon)}{C_2};\frac{L_{+}(C_1\varepsilon)}{C_2}]$, exept the ones which lie in $\Delta_0(\varepsilon)$.

So I'm interested in the correctness of the following hypothesis.

Hypothesis 1. Any non-constant trigonometric polynomial $f(t) = \sum\limits_{k=1}^{n}A_k e^{i\lambda_k t}, A_k \in \mathbb{C}, \lambda_k \in \mathbb{R}$, satisfies the property A.

It is easy to see that hypothesis 1 is true for the periodic case. But I even don't know what to do with special almost periodic case.

For the general case, i.e. $f(t) \sim \sum\limits_{k=1}^{\infty} A_k e^{i\lambda_k t}$ it seems that we have to put some conditions on the exponents $\lambda_k$, for example, make them to be separated from each other.

Answer to @fedja

Let's take a look at the periodic case. Consider the function $f(t)=\sin(t)$. In this case the set $\Omega_{\varepsilon}(f)$ is the union $\bigcup\limits_{k} \Delta_k(\varepsilon)$, where $\Delta_k(\varepsilon)$ is the small (for a sufficient small $\varepsilon$) interval around $2\pi k, \ k \in \mathbb{Z}.$ In this case $L_{+}(\varepsilon)=2\pi$. My question is can we separate the set $\Delta_0(\varepsilon)$ from the set $\Omega_{\varepsilon}(f) \setminus \Delta_0(\varepsilon)$ in terms of $L_{+}(\varepsilon)$, i.e. are there any constants $\varepsilon_0>0, C_1>0, C_2>0$ such that $[-\frac{L_{+}(C_1\varepsilon)}{C_2},\frac{L_{+}(C_1\varepsilon)}{C_2}] \cap \Omega_{\varepsilon}(f)=\Delta_0(\varepsilon) \ \ \forall \varepsilon \in (0,\varepsilon_0)$, i.e. the distance between $\Delta_0(\varepsilon)$ and $\Omega_{\varepsilon}(f) \setminus \Delta_0(\varepsilon)$ is not less than $\approx\frac{L_{+}(C_1\varepsilon)}{C_2}$ for all sufficient small $\varepsilon$. It is clear that for our case we can choose $C_1=1,C_2=2$ and $\varepsilon_0$ to be sufficient small.

I want such bound for the almost periodic case, or express the possibility of such bound in terms of the exponents $\lambda_k$.

As far as I have understood, you've just shown that $L_{+}(\frac{1}{q^2})$ is not less than $q^3$. I have a proof that $L_{+}(\varepsilon) \geq (\frac{1}{\varepsilon})^{n-1+o(1)}$, where $f(t)=\sum_{k=1}^{n}A_ke^{i \lambda_k t}$ is a trigonometric polynomial with rationally independent exponents. Your example shows that the upper bound of $L_{+}(\varepsilon)$ can have different asymptotic, depending on the exponents. For an almost periodic functions with the property A the upper and lower bounds have the same asymptotics, i.e. $L_{+}(\varepsilon) = (\frac{1}{\varepsilon})^{n-1+o(1)}$.

Can we express the property A in the terms of the exponents, or just give an example of an almost periodic function with the property A?

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    $\begingroup$ In my example I've shown a little bit more: namely that also $q$ is an $\varepsilon=1/q^4$-period that is neither in $\Delta_0(\varepsilon)$, nor separated from $0$ by $L_+(q\varepsilon)/q\ge q^2$. Am I missing something? $\endgroup$ – fedja Aug 25 '16 at 12:19
  • $\begingroup$ @fedja, oh sorry, I've missed it. So you answered the question about hypothesis 1. Thank you very much. I'm not good in the number theory. I'm studying these propeties of almost periodic functions by using dimension theory and it is very interesting what we can investigate a such properties via methods of dimension theory. I will study these diophantine approximations methods. But if you can tell me something about the last question it will be very useful for me $\endgroup$ – demolishka Aug 25 '16 at 14:03
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If I deciphered the definitions right, you want to show that if $\omega>0$ is an $\varepsilon$ period of $f$, then either every $\omega'\in[-\omega,\omega]$ is, or every interval of length $C\omega$ contains a $C\varepsilon$ period of $f$. Unfortunately, this is false even for $e^{2\pi it}+e^{2\pi i\lambda t}$ if $\lambda$ is chosen appropriately. Just look at the way the line winds on the torus. If $\lambda$ can be approximated by an irreducible fraction $p/q$ with precision $1/q^4$ but not twice better, say, $\omega=q$ is the first non-trivial time we come to the $1/q^4$- neighborhood of the "origin" and the previous winds of length $1$ are about $1/q$ apart. Now, when you consider the subsequent intervals of length $q$, the corresponding unit winds of the spiral start moving at the speed about $1/q^4$ sideways for every $q$ in time. So, after time $q^2$, we'll be away from the origin by $1/q^3$ on the "main wind of length $1$" (i.e., $\omega\in[q-0.5,q+0.5]$) and it won't happen until time about $q^3/3$ that any other wind will be able to come closer than by $1/q$, so you'll have a gap of length about $q^3$ without any $1/q^2$-periods. Now just take any irrational $\lambda$ that admits infinitely many such approximations.

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