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.
Highlights
The first-order theory of the class of finite fields was determined in a fundamental paper of Ax, [1]
Exponential sums in a model theoretic setting were discussed in [19], who noted that in positive characteristic the additive character is definable, and used equidistribution to determine the theory of certain reducts of pseudo-finite fields
A slightly different formulation of the same argument: it is clear that the additive group of an ultraproduct of finite fields with additive character admits a -definable subgroup such that the quotient is connected in the logic topology, namely the kernel of Ψ
Summary
The first-order theory of the class of finite fields was determined in a fundamental paper of Ax, [1]. We will see that in continuous logic, a natural theory does exist that answers to the above, paralleling closely Ax’s theorem of pseudo-finite fields. Decidability holds in a strong sense: given a sentence S (formed using the basic relations, connectives and quantifiers), and ǫ > 0, one can effectively find a sentence S′ such that |S − S′| < ǫ in any model of T , and a number field L, such that S′ has only quantifiers ranging L; and the set of possible values of S′ is F (Tn) for some explicit polynomial function F and some n. Exponential sums in a model theoretic setting were discussed in [19], who noted that in positive characteristic the additive character is definable, and used equidistribution to determine the theory of certain reducts of pseudo-finite fields. Zilber [23] used them with a view to quantum mechanical integrals, taking different limits than we do here
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