Fourier pairs of discrete support with little structure

TL;DR

This paper proves the existence of discrete support measures with Fourier transforms that cannot be generated by finite applications of the Poisson Summation Formula, highlighting complex structures in Fourier analysis.

Contribution

It introduces a simple proof demonstrating measures with discrete support whose Fourier transforms defy finite Poisson Summation constructions.

Findings

Existence of such measures with complex support structures

Fourier pairs not obtainable by finite Poisson Summation applications

Supports are not contained in finite unions of arithmetic progressions

Abstract

We give a simple proof of the fact that there exist measures on the real line of discrete support, whose Fourier Transform is also a measure of discrete support, yet this Fourier pair cannot be constructed by repeatedly applying the Poisson Summation Formula finitely many times. More specifically the support of both the measure and its Fourier Tranform are not contained in a finite union of arithmetic progressions.