Dynamics of multiplicative groups over fields and Folner-Kloosterman sums

TL;DR

This paper investigates the relationships between multiplicative groups of infinite fields, their actions on additive duals, and applies these insights to establish equidistribution results for F4lner-Kloosterman sums, revealing new behaviors and combinatorial consequences.

Contribution

It demonstrates that actions are disjoint unless fields are isomorphic and extends classical Kloosterman sums to infinite fields, uncovering new multiplicative asymmetries.

Findings

Actions are disjoint unless fields are isomorphic.

Established equidistribution of F4lner-Kloosterman sums over infinite fields.

Derived combinatorial results including sum-product phenomena and polynomial theorems.

Abstract

For two countably infinite fields whose multiplicative groups are isomorphic, we examine invariant couplings between the actions that these groups induce on the additive Pontryagin duals of the fields. We show that the actions are disjoint unless the fields themselves are isomorphic and the group isomorphism extends (possibly after a finite twist) to a field isomorphism. As an application, we establish equidistribution of F\o lner-Kloosterman sums - an extension of classical Kloosterman sums to infinite fields. Unlike the classical case over algebraic closures of finite fields, these averages exhibit an inherent multiplicative asymmetry, revealing new and fundamentally different behavior. Finally, we derive several combinatorial consequences, including results on sum-product phenomena and a Furstenberg--S\'ark\"ozy-type theorem for Laurent polynomials over general fields.