Sums of squares of integers from residue classes

A subset $\mathcal{A}\subseteq \mathbb{Z}$ is called $s$-almost square universal if every sufficiently large positive integer can be written as a sum of at most $s$ squares of integers from $\mathcal{A}$. In this article, we study the minimal number $\mathrm{A}\mathrm{S}\mathrm{U}({\mathcal{A}}_{d,m})$ with this property, where ${\mathcal{A}}_{d,m}$ denotes the residue class of $d$ modulo $m$, with $m\in \mathbb{N}$ and $d\in \mathbb{Z}$. We further prove that ${\mathcal{A}}_{d,m}$ is $s$-square universal for some $s\in \mathbb{N}$ if and only if $d\equiv \pm 1(\mathrm{m}\mathrm{o}\mathrm{d}m)$, and determine the minimal such number $\mathrm{S}\mathrm{U}({\mathcal{A}}_{d,m})$ in these cases.