Pseudofinite fields with additive and multiplicative character

We introduce the theory ${\mathrm{P}\mathrm{F}}^{+,\times }$ of pseudofinite fields with generic additive and multiplicative character added as continuous logic predicates. Using the Weil bounds on character sums over finite fields as well as the Erdős-Turàn-Koksma inequality we show that it is the asymptotic theory (in characteristic $0$) of finite fields with (sufficiently generic) additive and multiplicative character. Moreover, we establish quantifier elimination in a natural definitional expansion of the language and deduce that integration by the Chatzidakis-van den Dries-Macintyre counting measure is uniformly definable in the parameters. Finally, we show that ${\mathrm{P}\mathrm{F}}^{+,\times }$ is a simple theory.