Sums and Products of Distinct Sets and Distinct Elements in $\mathbb{C}$

TL;DR

This paper explores sum-product phenomena in complex sets, establishing lower bounds on sumsets and product sets under certain conditions, using advanced number theory and additive combinatorics techniques.

Contribution

It introduces a new sum-product inequality for complex sets with small product sets, combining results from multiplicative group equations and sumset theory.

Findings

If |AB|<α|A| and α ≪ log|A|, then |kA+lB| is significantly larger than |A|^k|B|^l.

Provides a lower bound on the sum of sumset and product set sizes for a set of complex numbers.

Extends sum-product phenomena to complex numbers using advanced algebraic tools.

Abstract

Let $A$ and $B$ be finite subsets of $\mathbb{C}$ such that $|B|=C|A|$. We show the following variant of the sum product phenomenon: If $|AB|<\alpha|A|$ and $\alpha \ll \log |A|$, then $|kA+lB|\gg |A|^k|B|^l$. This is an application of a result of Evertse, Schlickewei, and Schmidt on linear equations with variables taking values in multiplicative groups of finite rank, in combination with an earlier theorem of Ruzsa about sumsets in $\mathbb{R}^d$. As an application of the case $A=B$ we give a lower bound on $|A^+|+|A^\times|$, where $A^+$ is the set of sums of distinct elements of $A$ and $A^\times$ is the set of products of distinct elements of $A$.