Finding large additive and multiplicative Sidon sets in sets of integers

TL;DR

This paper demonstrates the existence of large additive and multiplicative Sidon sets within any finite set of integers, providing bounds that advance understanding of their size relative to the original set.

Contribution

It establishes new lower bounds for the size of large Sidon sets in any finite integer set, improving previous results and addressing conjectures in additive combinatorics.

Findings

Existence of constants g and δ ensuring large Sidon subsets within any finite set.

Quantitative bounds for the size of these subsets depending on the original set size.

Progress towards a conjecture of Klurman--Pohoata and improvements over prior work of Shkredov.

Abstract

Given $h,g \in \mathbb{N}$, we write a set $X \subset \mathbb{Z}$ to be a $B_{h}^{+}[g]$ set if for any $n \in \mathbb{Z}$, the number of solutions to the additive equation $n = x_1 + \dots + x_h$ with $x_1, \dots, x_h \in X$ is at most $g$, where we consider two such solutions to be the same if they differ only in the ordering of the summands. We define a multiplicative $B_{h}^{\times}[g]$ set analogously. In this paper, we prove, amongst other results, that there exist absolute constants $g \in \mathbb{N}$ and $\delta>0$ such that for any $h \in \mathbb{N}$ and for any finite set $A$ of integers, the largest $B_{h}^{+}[g]$ set $B$ inside $A$ and the largest $B_{h}^{\times}[g]$ set $C$ inside $A$ satisfy \[ \max \{ |B| , |C| \} \gg_{h} |A|^{(1+ \delta)/h }. \] In fact, when $h=2$, we may set $g = 31$, and when $h$ is sufficiently large, we may set $g = 1$ and $\delta \gg (\log \log h)^{1/2 - o(1)}$. The former makes progress towards a recent conjecture of Klurman--Pohoata and quantitatively strengthens previous work of Shkredov.