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 farther 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 as to approximations of
various types of Fourier transforms on them by the discrete Fourier
transform. Combining some methods of nonstandard analysis and
additive combinatorics we prove the three Gordon's Conjectures
which were open since 1991 and are crucial both in the formulations
and proofs of the LCA groups and Fourier transform approximation
theorems