Ruzsa's problem on Bi-Sidon sets

TL;DR

This paper investigates bi-Sidon sets, which are sets with distinct pairwise sums and products, and improves the lower bound on the size of such subsets within larger sets from a cubic root to a slightly larger exponent.

Contribution

The authors improve the lower bound on the size of bi-Sidon subsets contained in larger sets from $cN^{1/3}$ to $N^{1/3 + 7/78 + o(1)}$, advancing understanding of their structure.

Findings

Improved the lower bound on bi-Sidon subset size within larger sets.

Established a new exponent slightly larger than one-third for the subset size.

Enhanced previous results by Imre Ruzsa on bi-Sidon sets.

Abstract

A subset $S$ of real numbers is called bi-Sidon if it is a Sidon set with respect to both addition and multiplication, i.e., if all pairwise sums and all pairwise products of elements of $S$ are distinct. Imre Ruzsa asked the following question: What is the maximum number $f(N)$ such that every set $S$ of $N$ real numbers contains a bi-Sidon subset of size at least $f(N)$? He proved that $f(N)\geq cN^{\frac13}$, for a constant $c>0$. In this note, we improve this bound to $N^{\frac13+\frac7{78}+o(1)}$.