Asymmetric uniqueness sets in ${\ell }^q$

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.

26 pages