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 $\mathrm{ASU}(\mathcal{A}_{d,m})$ 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 $d \equiv \pm 1 \pmod{m}$, and determine the minimal such number $\mathrm{SU}(\mathcal{A}_{d,m})$ in these cases.