The Range of Rademacher Series in $${\mathbb R}^d$$ R d

Abstract

In this paper, we study Rademacher series with d-dimensional vector-valued coefficients. We first employ a new combinatorial technique to present a sufficient condition for the Rademacher range of a sequence with a unique direction equal to $${\mathbb R}^2$$ . This result also gives a positive answer to the question that whether the Rademacher range of $$\{(n^{-1},n^{-1}\ln ^{-1}(n+1))\}$$ is $${\mathbb R}^2$$ . Next, by constructing homogeneous Cantor sets, we prove that, for each $$s\in [1,d]$$ , there exists a sequence with a unique direction such that its Rademacher range of Hausdorff dimension s is dense in $${\mathbb R}^d$$ but not equal to $${\mathbb R}^d$$ .