The Bose-Chowla argument for Sidon sets

Abstract

Let h≥2 and let A=(A1,…,Ah) be an h-tuple of sets of integers. For nonzero integers c1,…,ch, consider the linear form φ=c1x1+c2x2+⋯+chxh. The representation functionRA,φ(n) counts the number of h-tuples (a1,…,ah)∈A1×⋯×Ah such that φ(a1,…,ah)=n. The h-tuple A is a φ-Sidon system of multiplicity g if RA,φ(n)≤g for all n∈Z. For every positive integer g, let Fφ,g(n) denote the largest integer q such that there exists a φ-Sidon system A=(A1,…,Ah) of multiplicity g withAi⊆[1,n]and|Ai|=q for all i=1,…,h. It is proved that, for all linear forms φ,lim supn→∞Fφ,g(n)n1/h<∞ and, for linear forms φ whose coefficients ci satisfy a certain divisibility condition,lim infn→∞Fφ,h!(n)n1/h≥1.