  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.
