# 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
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