Dynamics of multiplicative groups over fields and Folner-Kloosterman sums

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.