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 $E\subset \hat{G}$ that is strongly small-2 but not Riesz. More precisely, $|\alpha |\ast |\beta |$ 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 $\hat{G}$ 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.

arXiv comment22 pages. Substantially revised and extended version; the main theorem now covers every infinite compact abelian group