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

We exhibit an asymmetry phenomenon for uniqueness sets in ${\mathrm{\ell }}^q$. Specifically, we construct sets that do not support measures with ${\mathrm{\ell }}^q$-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 ${\mathrm{\ell }}^q$ uniqueness problems.