Asymmetric uniqueness sets in $\ell^q$

TL;DR

This paper reveals an asymmetry in Fourier uniqueness sets in l^q, showing that certain sets support measures with very different Fourier decay properties, highlighting a divergence between unilateral and bilateral l^q uniqueness problems.

Contribution

The authors construct sets demonstrating a fundamental asymmetry in l^q Fourier uniqueness, contrasting support for measures with l^q-summable coefficients and those with faster decay.

Findings

Sets exist that support measures with faster-than-polynomial decay

Such sets do not support measures with l^q-summable Fourier coefficients

Highlights divergence between unilateral and bilateral l^q uniqueness

Abstract

We exhibit an asymmetry phenomenon for uniqueness sets in $\ell^q$. Specifically, we construct sets that do not support measures with $\ell^q$-summable Fourier coefficients, yet simultaneously support measures whose positive frequencies decay faster than polynomials. In the language of Fourier uniqueness, this highlights a striking divergence between the unilateral and bilateral $\ell^q$ uniqueness problems.