Generalized Sidon sets

TL;DR

This paper provides precise asymptotic estimates for the size of residue class sets with bounded representation functions and extends these results to integer sets, addressing a longstanding question by Simon Sidon.

Contribution

It introduces sharp asymptotic bounds for generalized Sidon sets and applies these to solve an open problem related to sets of integers.

Findings

Asymptotic estimates for residue class sets

Extension of results to integer sets

Resolution of an old question of Simon Sidon

Abstract

We give asymptotic sharp estimates for the cardinality of a set of residue classes with the property that the representation function is bounded by a prescribed number. We then use this to obtain an analogous result for sets of integers, answering an old question of Simon Sidon.