This paper is concerned with the Diophantine properties of the sequence {ξθ~n},where 1≤ξ<θ and θ 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 μ_λ with λ = θ^(-1) as the uniform contractive ratio is logarithmic. This generalizes some results of Kershner and Bufetov-Solomyak,who consider the case of Bernoulli convolutions. As an application,we prove that μ_λ almost every x is normal to any base b ≥ 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.
