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arXiv · 2511.20243

Pseudofinite fields with additive and multiplicative character

Stefan Marian Ludwig
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Abstract

We introduce the theory $\mathrm{PF}^{+,\times}$ 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 Erd\H{o}s-Tur\`an-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 $\mathrm{PF}^{+,\times}$ is a simple theory.

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Stefan Marian Ludwig. 2025-11-25. Pseudofinite fields with additive and multiplicative character. https://arxiv.org/abs/2511.20243

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