This paper is concerned with the Diophantine properties of the sequence {ξθn}, 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 μλ 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 μλ ahaost 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.
We are concerned with the sets of quasi generic points in finite symbolic space. We estimate the sizes of the sets by the Billingsley dimension defined by Gibbs measures. A dimension formula of such set is given, which generalizes Bowen＇s result. An application is given to the level sets of Birkhoff average.
