Sets of absolute convergence of double trigonometric series

We obtain a sufficient condition for a set of plane measure zero to be a set of absolute convergence (an A.C.-set) for a double trigonometric series. Specifically, let $y=f(x)$ ($0\leqslant x\leqslant2\pi$) be a smooth curve and let $\bigvee\limits_0^{2\pi}\ln|f'(x)|<\infty$. Then, any set of positive linear measure lying on this curve is an A.C.-set.