On products of shifts in arbitrary fields

Abstract

We adapt the approach of Rudnev, Shakan, and Shkredov presented in [2] to prove that in an arbitrary field F, for all A ⊂ F finite with |A| < p 1/4 if p := Char(F) is positive, we have |A(A + 1)| |A| 11/9 , |AA| + |(A + 1)(A + 1)| |A| 11/9 .This improves upon the exponent of 6/5 given by an incidence theorem of Stevens and de Zeeuw.

Attaining the exponent $5/4$ for the sum-product problem in finite fields

“…This lemma unifies the ad hoc regularisation techniques present in the sum-product literature, e.g. [16,29]; an asymmetric formulation is recorded by Stevens and Warren [26]. Although Xue states this lemma over R, its proof is valid over abelian groups; similarly we may take k > 0 (see e.g.…”

On Growth of the Set $A(A+1)$ in Arbitrary Finite Fields

“…Warren [11], further improved this bound to (log |A|) −7/6 |A| 1+2/9 under the constraint |A| p 1/4 . Both of these results are based on a bound on incidences between lines and Cartesian products, proved in [9], which in turn relies on a bound on incidences between points and planes due to Rudnev [8].…”