Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8904455 | Acta Mathematica Scientia | 2017 | 12 Pages |
Abstract
This paper is concerned with the Diophantine properties of the sequence
{ξθn}, where
1â¤Î¾<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 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.
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Xiang GAO, Jihua MA,