Sums of squares of integers from residue classes

A subset [Formula: see text] is called [Formula: see text]-almost square universal if every sufficiently large positive integer can be written as a sum of at most [Formula: see text] squares of integers from [Formula: see text]. In this paper, we study the minimal number [Formula: see text] with this property, where [Formula: see text] denotes the residue class of [Formula: see text] modulo [Formula: see text], with [Formula: see text] and [Formula: see text]. We further prove that [Formula: see text] is [Formula: see text]-square universal for some [Formula: see text] if and only if [Formula: see text], and determine the minimal such number [Formula: see text] in these cases.