Bohr almost periodic function

Remarks

Every Bohr almost periodic function is bounded and uniformly continuous. Bochner proved that $f$ is Bohr almost periodic if and only if every sequence of translates of $f$ has a uniformly convergent subsequence. So the definition agrees with the general almost periodic function on the group $\R$.

Example: let $f$ be continuous with period $p > 0$. Every multiple $kp$, $k \in \Z$, satisfies $f(x + kp) = f(x)$, so it is an $\varepsilon$-almost period for every $\varepsilon > 0$. Every closed interval of length $p$ contains a multiple of $p$, so $f$ is Bohr almost periodic.

Non-example: $f(x) = \arctan x$ is continuous, but not Bohr almost periodic. For $\tau > 0$ the difference $\arctan(x + \tau) - \arctan x$ is largest at $x = -\tau/2$, where it equals $2 \arctan(\tau/2)$, and the case $\tau < 0$ is symmetric. So for $0 < \varepsilon < \pi$ the $\varepsilon$-almost periods are the $\tau$ with $\abs{\tau} \le 2\tan(\varepsilon/2)$. This set is bounded, hence not relatively dense.