Decomposition of subsets of finite fields

Abstract

We extend a bound of Roche-Newton, Shparlinski and Winterhof which says any subset of a finite field can be decomposed into two disjoint subset U and V of which the additive energy of U and f pVq are small, for suitably chosen rational functions f . We extend the result by proving equivalent results over multiplicative energy and the additive and multiplicative energy hybrids.

Low-energy decomposition results over finite fields

“…Our first result generalizes [13,Theorem 2.8]. This result also makes partial progress toward [11,Question 5.1] as it recovers an inequality of the form (8) where f may be taken to be any non-degenerate bivariate quadratic polynomial and g may be any degenerate function. Although, we fix g(x, y) = x + y for tidier presentation.…”

“…This estimate becomes non-trivial when |A| > q 1/2+o (1) and no similar results are known for smaller sets. Also see [8] for certain variations of [11,Theorem 1.1].…”

Exponential Sums and Additive Combinatorics

“…Our bounds rely on bounds on certain character sums. Our extensions will show that we can replace E + with E × in either or both terms in (1), as long as we suitably change our restriction on our function f. These results have been published in [4]. Multilinear exponential sums are those of the form…”