Vol. 8, No. 3, 2019

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On products of shifts in arbitrary fields

Audie Warren

Vol. 8 (2019), No. 3, 247–261

We adapt the approach of Rudnev, Shakan, and Shkredov (2018) to prove that in an arbitrary field F, for all A F finite with |A| < p14 if p := Char(F) is positive, we have

|A(A + 1)| |A|119 (log|A|)76,|AA| + |(A + 1)(A + 1)| |A|119 (log|A|)76.

This improves upon the exponent of 6 5 given by an incidence theorem of Stevens and de Zeeuw.

growth, sum-product estimates, energy
Mathematical Subject Classification 2010
Primary: 11B75, 68R05
Received: 6 December 2018
Revised: 30 April 2019
Accepted: 14 May 2019
Published: 23 July 2019
Audie Warren
Johann Radon Institute for Computational and Applied Mathematics