I was recently trying to make sense of the definition of Besicovitch almost periodicity. Motivated by the many equivalent definitions of Bohr almost periodicity, I was wondering whether the same holds for Besicovitch almost periodicity. I assume that all my functions are continuous and bounded, which I denote by $C_b$.
For a compact Abelian group $G$, we can define for $t\in G$ an operator $\tau_t:C_b(G)\to C_b(G)$ by
$$ [\tau_t f](g):= f\big( g-t \big) \quad \text{for all} \quad f\in C_b(G) \quad \text{and} \quad g\in G. $$
A Bohr almost-periodic function is a function in the uniform norm closure of finite linear combinations of characters or trigonometric polynomials. It is also equivalent to two other conditions.
The set of $\epsilon$ almost periods of $f$ is defined as
$$ \mathfrak{ap}_\epsilon(f):=\{ t\in G: \Vert f- \tau_t(f) \Vert_\infty<\epsilon \}. $$
$f$ is Bohr-almost periodic if and only if for all $\epsilon>0$, there exists a compact subset $K\subseteq G$ such that $G = \cup_{t\in \mathfrak{ap_\epsilon}}(t+K)$.
Also, $f$ is Bohr-almost periodic if and only if $\{ \tau_t(f): t\in G \}$ is relatively compact in $\big( C_b(G), \Vert \cdot \Vert_\infty \big)$.
For Besicovitch almost periodicty on amenable groups, one can define the Besicovitch semi-norm for a Van-Hove sequence $\mathcal{A}=\{ A_n \}$ by
$$ \Vert f\Vert_{Bes}= \underset{n\to \infty}{\limsup} \frac{1}{m_G(A_n)} \int_{A_n} \vert f \vert dm_G, $$
where $m_G$ is the Haar measure on $G$. We then say $f\in C_b(G)$ is Besicovitch almost-periodic if it is the Besicovitch seminorm closure of linear combinations of characters. Are there equivalent formulations like in the Bohr almost periodic case? Something like:
The following are equivalent:
$f$ is Besicovitch almost periodic
$\{ \tau_t(f): t\in G \}$ is relatively compact in $\big( C_b(G), \Vert \cdot \Vert_{Bes} \big)$.
For any $\epsilon>0$ there exists a compact $K$ such that $G= \cup_{t \in \mathfrak{ap}_\epsilon^{Bes}(f) } \big( t+K \big)$, where
$$ \mathfrak{ap}_\epsilon^{Bes}(f):=\{ t\in G: \Vert f- \tau_t(f) \Vert_{Bes}<\epsilon \}. $$
I saw the equivalence between the first and second point in the question on this thread in math overflow, but I haven't found this sort of statement elsewhere so I was wondering whether it is true. I'm not well versed with these notions, but I thought this might be well known to people working with Besicovitch almost periodic functions.
