New sum-product estimates for real and complex numbers

TL;DR

This paper establishes new lower bounds on the size of sets formed by additive and multiplicative combinations of finite sets of real and complex numbers, advancing sum-product theory.

Contribution

It proves novel sum-product estimates for finite sets of real and complex numbers, including explicit bounds on the size of certain combined sets.

Findings

For positive real sets, the set of ratios (a+b)/(c+d) has size at least 2|A|^2 - 1.

The k-fold sumset of A grows at least as fast as |A|^k.

Bounds also hold for complex sets with smaller constants.

Abstract

A variation on the sum-product problem seeks to show that a set which is defined by additive and multiplicative operations will always be large. In this paper, we prove new results of this type. In particular, we show that for any finite set $A$ of positive real numbers, it is true that $$\left|\left\{\frac{a+b}{c+d}:a,b,c,d\in{A}\right\}\right|\geq{2|A|^2-1}.$$ As a consequence of this result, it is also established that $$|4^{k-1}A^{(k)}|:=|\underbrace{\underbrace{A\cdots{A}}_\textrm{k times}+\cdots{+A\cdots{A}}}_\textrm{$4^{k-1}$ times}|\geq{|A|^k}.$$ Later on, it is shown that both of these bounds hold in the case when $A$ is a finite set of complex numbers, although with smaller multiplicative constants.