Exponential sums over definable subsets of finite fields

Details Exponential sums over definable subsets of finite fields Exponential sums over definable subsets of finite fields

2005-04-15

arxiv.org/abs/math/0504316v2

Mathematics

Number Theory

21 pages; correct statement of Theorem 1

Scientific paper

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.

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