On Products of Shifts in Arbitrary Fields

Abstract

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 $p:=Char(\mathbb{F})$ is positive, we have $$|A(A+1)|\gtrsim |A{|}^{11/9},|AA|+|(A+1)(A+1)|\gtrsim |A{|}^{11/9}.$$ This improves upon the exponent of $6/5$ given by an incidence theorem of Stevens and de Zeeuw.

References in corpus (4)

• An Improved Point-Line Incidence Bound Over Arbitrary Fields
• On higher energy decompositions and the sum-product phenomenon
• Breaking the 6/5 threshold for sums and products modulo a prime
• A new sum-product estimate in prime fields