On products of shifts in arbitrary fields

We adapt the approach of Rudnev, Shakan, and Shkredov (2018) to prove that in an arbitrary field $F$ , for all $A \subset F$ finite with $| A | < p^{1 ∕ 4}$ if $p : = Char (F)$ is positive, we have

| A (A + 1) | ≫ \frac{| A |^{11 ∕ 9}}{{(log | A |)}^{7 ∕ 6}}, | A A | + | (A + 1) (A + 1) | ≫ \frac{| A |^{11 ∕ 9}}{{(log | A |)}^{7 ∕ 6}} .

This improves upon the exponent of $\frac{6}{5}$ given by an incidence theorem of Stevens and de Zeeuw.

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Keywords

growth, sum-product estimates, energy

Mathematical Subject Classification 2010

Primary: 11B75, 68R05

Milestones

Received: 6 December 2018

Revised: 30 April 2019

Accepted: 14 May 2019

Published: 23 July 2019

Authors

Audie Warren
Johann Radon Institute for Computational and Applied Mathematics Linz Austria

Vol. 8, No. 3, 2019

Audie Warren

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors