On Uniqueness Properties of Rademacher Chaos Series

For a given integer $l\ge 1$, let $\{m_k\}$ be an arbitrary numeration of the integers permitting a dyadic decomposition $2^{k_1}+2^{k_2}+\ldots+2^{k_s}$ with $s\le l$. We prove that (i) the convergence of a Walsh series $\sum_ka_kw_{m_k}(x)$ on a set of measure $>1-2^{-4l}$ implies $\sum_ka_k^2<\infty$ and (ii) if it converges to zero on a set of the same measure $>1-2^{-4l}$, then $a_k=0$ for all $k\ge 1$.