Asymptotically, the solutions of Waring’s problem follow a limit law of which we are able to compute explicitly the limit density. In the special cases of sums of 3 and 4 squares where such a result is not possible, we establish a distribution result for slices of at least h0(n) consecutive integers ending at n, that is integers from n−h0(n)+1 to n, where h0(n) = nε for 4 squares and h0(n) = n¼+ε for 3 squares (ε > 0). We then deduce from this study the asymptotic behaviour of some kind of Riemann sums with an arithmetic constraint for which we point out an application related to the study of Schrödinger equation.