Strongly small-2 sets which are not Riesz sets

Abstract

We construct spectral sets in discrete abelian groups which are strongly small-2 but are not Riesz sets. More precisely, on each of the compact groups [ (\mathbb Z/6\mathbb Z)^{\mathbb N} \qquad\text{and}\qquad (\mathbb Z/p\mathbb Z)^{\mathbb N}, ] where (p) is an odd prime, there is a proper spectral set (E) such that [ |α|*|β| \ll m ] for every (α,β\in M_E(G)), although (M_E(G)) contains a nonzero singular measure. The latter may be chosen with mass one, Fourier support exactly (E), and convolution square in (L^2(G)). The construction combines an elementary finite-intersection criterion with a summable perturbation of a singular positive product measure. In the first model, the same full spectral support admits an amplitude family with an exact (\ell^2) absolute-continuity/singularity dichotomy. It also yields an uncountable family of mutually singular measures and an isometric copy of [ \ell^1\bigl((0,1/2]\bigr) ] in the square-zero quotient (M_E(G)/L^1_E(G)).