Sum-product estimates over arbitrary finite fields

TL;DR

This paper establishes new sum-product estimates over finite fields, demonstrating that small sets exhibit significant growth in sum and product combinations, advancing understanding of additive and multiplicative structures in finite fields.

Contribution

It provides novel sum-product bounds over arbitrary finite fields, extending Erdős distance problem analogues to these algebraic structures.

Findings

For small sets A, |(A-A)^2+(A-A)^2| grows faster than |A|^{1+1/21}.

Max of |A+A| and |A^2+A^2| exceeds |A|^{1+1/42}.

|A+A^2| exceeds |A|^{1+1/84}.

Abstract

In this paper we prove some results on sum-product estimates over arbitrary finite fields. More precisely, we show that for sufficiently small sets $A\subset \mathbb{F}_q$ we have \[|(A-A)^2+(A-A)^2|\gg |A|^{1+\frac{1}{21}}.\] This can be viewed as the Erd\H{o}s distinct distances problem for Cartesian product sets over arbitrary finite fields. We also prove that \[\max\{|A+A|, |A^2+A^2|\}\gg |A|^{1+\frac{1}{42}}, ~|A+A^2|\gg |A|^{1+\frac{1}{84}}.\]