Multiple Walsh Series and Zygmund Sets

The classical Zygmund theorem claims that, for any sequence of positive numbers $\{\varepsilon_n\}$ monotonically tending to zero and any $\delta>0$, there exists a set of uniqueness for the class of trigonometric series whose coefficients are majorized by the sequence $\{\varepsilon_n\}$ whose measure is greater than $2\pi-\delta$. In this paper, we prove the analog of Zygmund's theorem for multiple series in the Walsh system on whose coefficients rather weak constraints are imposed.