An application of linear programming duality to discrete Fourier analysis and additive problems

TL;DR

This paper explores how linear programming duality, specifically the separating hyperplane principle, can be applied to discrete Fourier analysis and additive problems in number theory, providing new insights even when direct perturbations of functions are not possible.

Contribution

It introduces a novel application of linear programming duality to additive number theory, connecting Fourier analysis with optimization techniques.

Findings

Utilizes the separating hyperplane principle in the context of Fourier analysis.

Provides a method to derive properties of functions on Z_p using linear programming.

Offers new perspectives on additive problems through duality arguments.

Abstract

Suppose that f is a function from Z_p -> [0,1] (Z_p is my notation for the integers mod p, not the p-adics), and suppose that a_1,...,a_k are some places in Z_p. In some additive number theory applications it would be nice to perturb f slightly so that Fourier transform f^ vanishes at a_1,...,a_k, while additive properties are left intact. In the present paper, we show that even if we are unsuccessful in this, we can at least say something interesting by using the principle of the separating hyperplane, a basic ingredient in linear programming duality.