On sets of absolute convergence for multiple trigonometric series

We obtain a sufficient condition for a set of measure zero in $N$-dimensional space to be a set of absolute convergence (A. C. set) for an $N$-tuple trigonometric series. We also show that, in a certain subclass of sets of measure zero (namely in the subclass of ldquo monotonicrdquo curves), this condition cannot be sharpened.