Sets of uniqueness for spherically convergent multiple trigonometric series

A subset $E$ of the $d$-dimensional torus $\mathbb{T} ^{d}$ is called a set of uniqueness, or $U$-set, if every multiple trigonometric series spherically converging to $0$ outside $E$ vanishes identically. We show that all countable sets are $U$-sets and also that $H^{J}$ sets are $U$-sets for every $J$. In particular, $C\times\mathbb{T} ^{d-1}$, where $C$ is the Cantor set, is an $H^{1}$ set and hence a $U$-set. We will say that $E$ is a $U_{A}$-set if every multiple trigonometric series spherically Abel summable to $0$ outside $E$ and having certain growth restrictions on its coefficients vanishes identically. The above-mentioned results hold also for $U_{A}$ sets. In addition, every $U_{A}$-set has measure $0$, and a countable union of closed $U_{A}$-sets is a $U_{A}$-set.