Infinite sumsets with many representations

TL;DR

This paper investigates the structure of infinite sets of nonnegative integers whose sumsets have many representations, establishing a lower bound on the counting function based on the number of representations.

Contribution

It provides a new lower bound on the growth of the set based on the multiplicity of representations in sumsets for large integers.

Findings

Sets with many representations in sumsets grow at least as fast as (log x)/log h

If large integers in hA have multiple representations, then A(x) is bounded below by a logarithmic function

The result links the number of representations to the density of the original set A.

Abstract

Let $A$ be an infinite set of nonnegative integers. For $h \geq 2$, let $hA$ be the set of all sums of $h$ not necessarily distinct elements of $A$. If every sufficiently large integer in the sumset $hA$ has at least two representations, then $A(x) \geq (\log x)/\log h)-w_0$, where $A(x)$ counts the number of integers $a \in A$ such that $1 \leq a \leq x$.