Extremal properties of product sets

TL;DR

This paper determines the nearly optimal size of subsets of [N] for which the product set has specific extremal sizes, addressing open problems in additive combinatorics.

Contribution

It provides nearly optimal bounds for the size of sets with extremal product set properties, resolving previously posed open questions.

Findings

Established bounds for |A| when |AA| is approximately |A|^2/2.

Determined the size of A for |AA| close to |[N][N]|.

Solved open problems from Cilleruelo, Ramana, and Ramare.

Abstract

We find nearly the optimal size of a set $A\subset [N] := \{1,...,N\}$ so that the product set $AA$ satisfies either (i) $|AA| \sim |A|^2/2$ or (ii) $|AA| \sim |[N][N]|$. This settles problems raised in a recent article of Cilleruelo, Ramana and Ramare.