Lagrange’s theorem with 𝑁^{1/3} squares

Abstract

For every N > 1 N > 1 we construct a set A of squares such that | A | > ( 4 / log ⁡ 2 ) N 1 / 3 log ⁡ N |A| > (4/\log 2){N^{1/3}} \log N and every nonnegative integer n ⩽ N n \leqslant N is a sum of four squares belonging to A.