Pseudofinite difference fields and counting dimensions

TL;DR

This paper investigates ultraproducts of finite fields with Frobenius automorphism, revealing their model-theoretic properties and exploring the relationship between coarse dimension and transformal transcendence degree.

Contribution

It introduces a family of difference fields with definable, integer-valued coarse pseudofinite dimension and analyzes their model-theoretic complexity.

Findings

Theories have the strict order property and TP2.

Coarse pseudofinite dimension is definable and integer-valued.

Potential link between coarse dimension and transformal transcendence degree.

Abstract

We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we also discuss the possible connection between coarse dimension and transformal transcendence degree in these difference fields.