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 uniqu...
