# Stronger sum-product inequalities for small sets

@article{Rudnev2018StrongerSI,
title={Stronger sum-product inequalities for small sets},
author={Misha Rudnev and George Shakan and Ilya D. Shkredov},
journal={arXiv: Combinatorics},
year={2018}
}
• Published 25 August 2018
• Mathematics
• arXiv: Combinatorics
Let $F$ be a field and a finite $A\subset F$ be sufficiently small in terms of the characteristic $p$ of $F$ if $p>0$. We strengthen the "threshold" sum-product inequality $$|AA|^3 |A\pm A|^2 \gg |A|^6\,,\;\;\;\;\mbox{hence} \;\; \;\;|AA|+|A+A|\gg |A|^{1+\frac{1}{5}},$$ due to Roche-Newton, Rudnev and Shkredov, to $$|AA|^5 |A\pm A|^4 \gg |A|^{11-o(1)}\,,\;\;\;\;\mbox{hence} \;\; \;\;|AA|+|A\pm A|\gg |A|^{1+\frac{2}{9}-o(1)},$$ as well as $$|AA|^{36}|A-A|^{24} \gg |A|^{73-o(1)}.$$ The latter… Expand
Sum-Product Type Estimates over Finite Fields
Let $\mathbb{F}_q$ denote the finite field with $q$ elements where $q=p^l$ is a prime power. Using Fourier analytic tools with a third moment method, we obtain sum-product type estimates for subsetsExpand
A note on sum-product estimates over finite valuation rings
Let $\mathcal R$ be a finite valuation ring of order $q^r$ with $q$ a power of an odd prime number, and $\mathcal A$ be a set in $\mathcal R$. In this paper, we improve a recent result due to YaziciExpand
A NEW SUM–PRODUCT ESTIMATE IN PRIME FIELDS
• Mathematics
• Bulletin of the Australian Mathematical Society
• 2019
We obtain a new sum–product estimate in prime fields for sets of large cardinality. In particular, we show that if $A\subseteq \mathbb{F}_{p}$ satisfies $|A|\leq p^{64/117}$ then $\max \{|A\pmExpand An update on the sum-product problem • Mathematics • Mathematical Proceedings of the Cambridge Philosophical Society • 2021 We improve the best known sum-product estimates over the reals. We prove that $\max(|A+A|,|A+A|)\geq |A|^{\frac{4}{3} + \frac{2}{1167} - o(1)}\,,$ for a finite$A\subset \mathbbExpand
A new perspective on the distance problem over prime fields.
• Mathematics
• 2019
Let $\mathbb{F}_p$ be a prime field, and ${\mathcal E}$ a set in $\mathbb{F}_p^2$. Let $\Delta({\mathcal E})=\{||x-y||: x,y \in {\mathcal E} \}$, the distance set of ${\mathcal E}$. In this paper, weExpand
A note on conditional expanders over prime fields