Almost periodic functions and the property A

Question

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?

Answer 1

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.