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

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 $A$.

27 pages, Revised with corrections. Accepted by Integers: Electronic Journal of Combinatorial Number Theory