Gordon’s Conjectures 1 and 2: Pontryagin–van Kampen duality in the hyperfinite setting

Abstract

Using the ideas of E. I. Gordon we present and further advance an approach, based on nonstandard analysis, to simultaneous approximations of locally compact abelian groups and their duals by (hyper)finite abelian groups, as well of the Haar measures on them. Combining the techniques of nonstandard analysis and the Fourier analytic methods of additive combinatorics we prove the first two of the three Gordon’s Conjectures which were open since 1991 and are crucial both in the formulations and proofs of the approximation theorems for LCA groups and for the Fourier transform.