  For $0<\alpha\le 1$, we say that a sequence $(X_k)_{k>0}$ of $d$-regular
graphs has property $D_\alpha$ if there exists a constant $C>0$ such that
$\mathrm{diam}(X_k)\ge C\cdot|X_k|^\alpha$. We investigate property $D_\alpha$
for arithmetic box spaces of the solvable Baumslag-Solitar groups $BS(1,m)$
(with $m\geq 2$): those are box spaces obtained by embedding $BS(1,m)$ into the
upper triangular matrices in $GL_2(\mathbb{Z}[1/m])$ and intersecting with a
family $M_{N_k}$ of congruence subgroups of $GL_2(\mathbb{Z}[1/m])$, where the
levels $N_k$ are coprime with $m$ and $N_k|N_{k+1}$. We prove:
  - if an arithmetic box space has $D_\alpha$, then $\alpha\le\frac{1}{2}$~;
  - if the family $(N_k)_k$ of levels is supported on finitely many primes, the
corresponding arithmetic box space has $D_{1/2}$~;
  - if the family $(N_k)_k$ of levels is supported on a family of primes with
positive analytic primitive density, then the corresponding arithmetic box
space does not have $D_\alpha$, for every $\alpha>0$.
  Moreover, we prove that if we embed $BS(1,m)$ in the group of invertible
upper-triangular matrices $T_n(\mathbb{Z}[1/m])$, then every finite index
subgroup of the embedding contains a congruence subgroup. This is a version of
the congruence subgroup property (CSP).
