Ax's theorem with an additive character

TL;DR

This paper extends Ax's theorem to a setting involving additive characters, connecting model theory, exponential sums, and algebraic geometry over finite fields, with implications for definable measures and approximation of functions.

Contribution

It generalizes Ax's theorem to include additive characters and develops model-theoretic tools for analyzing exponential sums over finite fields.

Findings

Generalization of Ax's theorem with additive characters

Model-theoretic development of definable measures

Approximation of functions by polynomial expressions in additive characters

Abstract

Motivated by Emmanuel Kowalski's exponential sums over definable sets in finite fields, we generalize Ax's theorem on pseudo-finite fields to a continuous-logic setting allowing for an additive character. The role played by Weil's Riemann hypothesis for curves over finite fields is taken by the `Weil bound' on exponential sums. Subsequent model-theoretic developments, including simplicity and the Chatzidakis-Van den Dries-Macintyre definable measures, also generalize. Analytically, we have the following consequence: consider the algebra of functions $\Ff_p^n \to \Cc$ obtained from the additive characters and the characteristic functions of subvarieties by pre- or post-composing with polynomials, applying min and sup operators to the real part, and averaging over subvarieties. Then any element of this class can be approximated, uniformly in the variables and in the prime $p$, by a polynomial expression in $\Psi_p(\xi)$ at certain algebraic functions $\xi$ of the variables, where $\Psi(n \mod p) = exp(2 \pi i n/p)$ is the standard additive character.