Explicit multidimensional Ingham--Beurling type estimates

TL;DR

This paper presents a simplified, elementary proof of multidimensional Ingham--Beurling type estimates for nonharmonic Fourier series, improving explicit constants and extending the results to higher dimensions.

Contribution

It introduces a short, elementary method to prove multidimensional Ingham--Beurling estimates, replacing complex analytical techniques used previously.

Findings

Simplified proof method for multidimensional estimates

Explicit constants provided in the estimates

Extension of results to higher dimensions

Abstract

Recently a new proof was given for Beurling's Ingham type theorem on one-dimensional nonharmonic Fourier series, providing explicit constants. We improve this result by applying a short elementary method instead of the previous complex analytical approach. Our proof equally works in the multidimensional case.