On Uniqueness Properties of Rademacher Chaos Series

For a given integer l >= 1, let {m(k)} be an arbitrary numeration of the integers permittinga dyadic decomposition 2(k1)+2(k2)+...+2(ks) with s <= l. We prove that (i) the convergence ofa Walsh series & sum;(k)a(k)w(mk)(x)on a set of measure >1-2(-4l) implies & sum;(k)a(k)(2)<infinity and (ii) if itconverges to zero on a set of the same measure >1-2(-4l), then a(k)=0 for all k >= 1.