Four Squares with Few Squares

Abstract

The classic theorem of Lagrange states that every nonnegative integer n is the sum of four squares. How “sparse” can a set of squares be and still retain the four square property. For any set X of nonnegative integers set N_x(x) = |{i ∈ X, ≤x}|. Let S = {0,1,4,9,…} denote the squares. If $X \subseteq S$ and every n ≥ 0 can be expressed as the sum of four elements of X then how slow can be the growth rate of Nx(x) ? Clearly we must have = N(x^1/4). Our object here is to give a quick proof of the following result of Wirsing[3]