The Beurling-Selberg Box Minorant Problem via Linear Programming Bounds

TL;DR

This paper explores the problem of constructing Fourier-supported minorants for high-dimensional indicator functions, revealing dimension-dependent existence results and providing explicit constructions for low dimensions, with applications to exponential sum inequalities.

Contribution

It introduces a high-dimensional Beurling-Selberg minorant problem, establishes non-existence results in large dimensions, and constructs explicit minorants for dimensions up to five.

Findings

No positive-mass minorants exist in sufficiently high dimensions.

Explicit minorants are constructed for dimensions 1 to 5.

Improved diophantine inequalities for exponential sums are derived.

Abstract

In this paper we investigate a high dimensional version of Selberg's minorant problem for the indicator function of an interval. In particular, we study the corresponding problem of minorizing the indicator function of the box $Q_{N}=[-1,1]^N$ by a function whose Fourier transform is supported in the same box $Q_N$. We show that when the dimension is sufficiently large there are no minorants with positive mass and we give an explicit lower bound for such dimension. On the other hand, we explicitly construct minorants for dimensions $1,2,3,4$ and $5$ and, as an application, we use them to produce an improved diophantine inequality for exponential sums.