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.
Categories: math.NT