Gordon's Conjectures: Pontryagin-van Kampen duality and the Fourier transform in hyperfinite setting

TL;DR

This paper advances a nonstandard analysis approach to approximate locally compact abelian groups and their Fourier transforms using hyperfinite groups, successfully proving three long-standing conjectures from 1991.

Contribution

It introduces a novel nonstandard analysis framework for approximating LCA groups and Fourier transforms, proving Gordon's three conjectures.

Findings

Proved Gordon's three conjectures from 1991.

Established approximation methods for LCA groups and Fourier transforms.

Connected nonstandard analysis with additive combinatorics.

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.