Almost periodic function

Show that $f(x) = \cos 2\pi x +\cos 2\pi \sqrt x$ is almost-periodic by showing directly that given $\varepsilon > 0$ there exists an integer $M$ such that at least one of any $M$ consecutive integers lies within $\varepsilon$ from an integral multiple of $2\pi$.

A set $\mathcal\subset \mathbb$ is relatively dense in $\mathbb$ if there exists a constant $L>0$ such that $(x,x+L)\cap F \neq \emptyset$ for all $x\in \mathbb$. Then a bounded uniformly continuous function $f:\mathbb\longrightarrow \mathbb$ is "almost periodic" iff for any $\varepsilon >0$, the set $$ \mathcal_\varepsilon = \left\lbrace h\in \mathbb:\sup_ \mathbb> \left|f(x+h) - f(x)\right| <\varepsilon \right\rbrace$$ is relatively dense in $\mathbb$.

$ then $$ f(x+m) - f(x) = \cos(2\pi \sqrt x + 2\pi \sqrtm) - \cos (2\pi \sqrt x)$$ and thus $$ |f(x+m) - f(x)| \leq 2\pi\Big( \inf_> \left|m\sqrt+n\right| \Big)$$ We know that the set $\left\lbrace m\sqrt + n: m,n\in \mathbb\right\rbrace$ is dense in $\mathbb$ (by a simple Pidgeonhole argument). Thus the problem will be soved if we can show that for any $\varepsilon>0$, the set $$ \mathcal = \left\lbrace m\in \mathbb: \;\text\;n\in \mathbb\;\text\quad |m\sqrt+n| < \varepsilon\right\rbrace$$ is "relatively dense" in $\mathbb$, where relatively dense means there exists a constant $M_\varepsilon>0$ such that any interval $(x,x+M_\varepsilon)$ contains at least one element of $\mathcal$. This somehow relates to the problem of approximation $\sqrt$ by rational numbers $\frac

$ but we want the set of $q$ is relatively dense. One well-known result is the Dirichlet approximation theorem, which states that showing that any real number has a sequence of good rational approximations: in fact an immediate consequence is that for a given irrational $\alpha$, the inequality $$\left|\alpha - \frac

\right| < \frac$$ is satisfied by infinitely many integers $p$ and $q$. But the problem how to control the distance between two consecutive denumerators still cannot be solved. Can anyone help me? Thank you very much.

• real-analysis
• functional-analysis
• number-theory
• diophantine-approximation
• almost-periodic-functions