# Non uniform continuity of a function and almost periodicity

We say that a continuous function $f:\mathbb{R}\to\mathbb{C}$ is almost periodic in the sense of Bohr if:

For every sequence $(t'_n)_{n\geq0}$, there's a sub-sequence $(t_n)_{n\geq0}$ such that $f(t+t_n)$ converges uniformly in $\mathbb{R}$ to a function $g(t)$ i.e. $$\sup_{t\in \mathbb{R}}|f(t+t_n)-g(t)|\to 0, \ \ when \ \ n\to +\infty.$$

This class of functions was proved to be the closure under the uniform norm in $\mathbb{R}$ of the class of Trigonometric Polynomial with independent frequencies $$P(t)=\sum_{k=0}^N a_ke^{ib_kt}$$

Almost periodic functions are uniformly continuous and this class contains the class of periodic functions, but functions like $f(t)=\sin(t)+\sin(t\sqrt2)$ are almost periodic but not periodic. We can show this by assuming it is periodic then get the contradiction $\sqrt2 \in \mathbb{Q}$.

A more general class is the class the almost automorphic functions. A continuous function $f:\mathbb{R}\to\mathbb{C}$ is almost automorphic if

For every sequence $(t'_n)_{n\geq0}$, there's a sub-sequence $(t_n)_{n\geq0}$ and a function $g$ such that for each $t\in \mathbb{R}$ $$|f(t+t_n)-g(t)|\to 0, \ \ when \ \ n\to +\infty.$$ and $$|g(t-t_n)-f(t)|\to 0, \ \ when \ \ n\to +\infty.$$

This class is proved to contain the class of almost periodic functions. A classic example of almost automoprhic function which is not almost periodic is: $$f(t)=\sin\left(\frac{1}{2+\sin(t)+\sin(t\sqrt2)}\right)$$ To prove this, we only have to show that this function is not uniformly continuous(since almost periodic functions are uniformly continuous). But It seems like I cannot prove it. I also cannot prove that this function is almost automorphic using the definition.

• I think it would be useful if we find a sequence $t_n$ such that $2+\sin(t_n)+\sin(t_n \sqrt 2)$ goes to $0$, or in other words $\sin(t_n)$ and $\sin(t_n\sqrt 2)$ going to $-1$ at the same time. – user165633 May 10 '14 at 19:01

You must see (read) the following work (of Bolis Basit and Hans Günzler, posted at arxiv):

http://arxiv.org/abs/1006.2169

There you can find nice steps to prove the desired claims.

Best.

• Thank you, this is exactly what I am looking for !!! – user165633 Jul 20 '14 at 20:41

I'd suggest looking carefully at a graph of this function, really quite remarkable. I'd look at the area with "sin(1/x)" type behavior . The scale factor is 100 along the y-axis. (This is intended to be a comment, but it's necessary to look at some areas that are clearly not well behaved. )

• There's actually no $\sin(1/x)$ like area in the graph, those areas contains finite oscillations unlike $\sin(1/x)$. But the number of oscillations is not bounded as you move from "bad" area to "bad" area. I also saw that $2+\sin(t)+\sin(\sqrt 2 t)$ can never reach $0$ but there must be a sequence such that $2+\sin(t_n)+\sin(\sqrt 2 t_n)$ goes to $0$. It seems to me like this has to do with how we can approximate $\sqrt 2$ with rationals. – user165633 May 10 '14 at 18:38
• Yes , I see, very interesting example. – Alan May 10 '14 at 19:14
• It is a weird function :). – user165633 May 10 '14 at 19:20