Large $B_2[g]$ subsets of the first squares

Abstract

We prove that, for every fixed integer $g\ge 1$, the largest cardinality of a $B_2[g]$ subset of the first $n$ squares is at least a positive constant, depending only on $g$, times $$n^{\frac{2g}{2g+1}}(\log n{)}^{\frac{2-2^g}{2g+1}},$$ for all sufficiently large $n$. For $g=1$, this recovers the theorem of Lefmann and Thiele on Sidon subsets of the first squares. The proof follows their hypergraph method, but replaces the $4$-uniform hypergraph encoding two representations as a sum of two squares by a $2(g+1)$-uniform hypergraph encoding $g+1$ such representations. The main point is to verify that this higher-uniformity hypergraph has few edges and few $2$-cycles; the lower bound then follows from an independence theorem for uncrowded hypergraphs due to Duke, Lefmann and Rödl.