On the size of the set $AA+A$

TL;DR

This paper proves a lower bound on the size of the set AA+A for finite positive real sets, and provides a counterexample to a conjecture by Balog showing the set can be smaller than quadratic.

Contribution

It establishes a universal lower bound for |AA+A| and constructs a set with smaller than quadratic size, challenging previous conjectures.

Findings

|AA+A| has a lower bound of |A|^{3/2 + c} for some c>0.

Existence of sets with |AA+A| = o(|A|^2), disproving Balog's conjecture.

Provides explicit constructions and bounds related to sum-product phenomena.

Abstract

It is established that there exists an absolute constant $c>0$ such that for any finite set $A$ of positive real numbers $$|AA+A| \gg |A|^{\frac{3}{2}+c}.$$ On the other hand, we give an explicit construction of a finite set $A \subset \mathbb R$ such that $|AA+A|=o(|A|^2)$, disproving a conjecture of Balog.