Fourier Pairs of Discrete Support with Little Structure

We give a simple proof of the fact, first proved in a stronger form in Lev and Olevskii (Quasicrystals with discrete support and spectrum, arXiv preprint arXiv:1501.00085 , 2014 ), 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 Transform are not contained in a finite union of arithmetic progressions.