B2[g] Sets and a Conjecture of Schinzel and Schmidt

Abstract

A set of integers is called a B2[g] set if every integer m has at most g representations of the form m = a + a′, with a ≤ a′ and a, a′ ∈ . We obtain a new lower bound for F(g, n), the largest cardinality of a B2[g] set in {1,. . .,n}. More precisely, we prove that infn→∞$\frac{F(g, n)}{\sqrt{gn}}\geq \frac 2{\sqrt \pi}-\e_g$ where ϵg → 0 when g → ∞. We show a connection between this problem and another one discussed by Schinzel and Schmidt, which can be considered its continuous version.