Summary: This paper is concerned with the Diophantine properties of the sequence \(\left\{ {\xi {\theta ^n}} \right\}\), where \(1 \leq \xi < \theta \) and \(\theta\) is a rational or an algebraic integer. We establish a combinatorial proposition which can be used to study such two cases in the same manner. It is shown that the decay rate of the Fourier transforms of self-similar measures \({\mu_\lambda}\) with \(\lambda = {\theta ^{ - 1}}\) as the uniform contractive ratio is logarithmic. This generalizes some results in literatures, which consider the case of Bernoulli convolutions. As an application, we prove that for \({\mu_\lambda}\), almost every \(x\) is normal to any base \(b \geq 2\), which implies that there exist infinitely many absolute normal numbers on the corresponding self-similar set. This can be seen as a complementary result of the well-known Cassels-Schmidt theorem.