Hausdorff dimensions of level sets of Rademacher series

Abstract

Let { a n } n≥1 be the sequence of real numbers such that lim n→∞ a n=0 and ∑ n=1 ∞ |a n|=∞ . Consider the Rademacher series S(x)= ∑ n=1 ∞ a nR n(x) and its level sets E a ={ x∈[0,1]: S( x)= a} (for all a∈ R ). We prove that dim H E a=1 for any a∈ R , where dim H is Hausdorff dimension.