Exponential sums over definable subsets of finite fields

Abstract

Abstract. We prove some general estimates for exponential sums over subsets of finite fields which are definable in the language of rings. This generalizes both the classical exponential sum estimates over varieties over finite fields due to Weil, Deligne and others, and the result of Chatzidakis, van den Dries and Macintyre concerning the number of points of those definable sets. As a first application, there is no formula in the language of rings that defines for infinitely many primes an "interval" in Z/pZ that is neither bounded nor with bounded complement.

A survey of algebraic exponential sums and some applications

“…This leads to the following statement (see [Ko1,Th. 13], which is a bit more general and more precise).…”

“…So although we have not yet been able to find convincing purely arithmetic applications, we can prove the following statement (see [Ko1,Remark 19]), where equidistribution reappears:…”

“…This set is not the set of points of an algebraic variety, 7 but one may wonder if it is possible to extend the language used to define algebraic sets to allow for a richer spectrum of possibilities that might contain these "short intervals". This, and the encounter with the paper [CDM], led the author (see [Ko1]) to try to look at exponential sums over definable sets over finite fields. These are probably the simplest generalizations of algebraic varieties: they amount to replacing the set of points V (F p ) with a definable set ϕ(F p ) associated with a formula in the first order language of rings, i.e., the set of x ∈ F p for which the formula is satisfied.…”

“…These sums, to an analytic number theorist, are even worth investigating independently of possible applications, and this study was begun in [Ko1].…”

Three-term polynomial progressions in subsets of finite fields

“…Kowalski, Proposition 9.ii of[11]). Let V ⊂ A n Z be an affine subscheme and F, G ∈ Z[V ] be regular functions on V .…”