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}
}
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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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
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We adapt the approach of Rudnev, Shakan, and Shkredov to prove that in an arbitrary field $\mathbb{F}$, for all $A \subset \mathbb{F}$ finite with $|A| < p^{1/4}$ if $p:= Char(\mathbb{F})$ isExpand
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We prove a range of new sum-product type growth estimates over a general field $\mathbb{F}$, in particular the special case $\mathbb{F}=\mathbb{F}_p$. They are unified by the theme of "breaking theExpand
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