Strongly small-2 sets that are not Riesz sets on compact abelian groups

Abstract

We prove that every infinite compact abelian group $G$ admits a countably infinite proper set $E\subset\widehat G$ that is strongly small-2 but not Riesz. More precisely, $|\alpha|*|\beta|$ is absolutely continuous for all $\alpha,\beta\in M_E(G)$, whereas $M_E(G)$ contains a singular complex measure of total mass one with exact Fourier support $E$ whose convolution square belongs to $L^2(G)$. When the image of the doubling map on $\widehat G$ is infinite, $E$ may moreover be chosen so that $\#(E\cap(\gamma-E))<\infty$ for every $\gamma$ and $E\cap(-E)=\{0\}$. When that image is finite, we reduce the problem to a nonrectangular construction on the Walsh group, for which $E\cap(E+\gamma)$ is finite at every nonzero shift.