B 2 [ g ] Sets and a Conjecture of Schinzel and Schmidt

Source abstract

A set of integers is called a B 2 [ 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 B 2 [ g ] set in {1,. . ., n }. More precisely, we prove that inf n →∞ $\frac{F(g, n)}{\sqrt{gn}}\geq \frac 2{\sqrt \pi}-\e_g$ where ϵ g → 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.