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Decay rate of Fourier transforms of some self-similar measures
- Source :
- Acta Mathematica Scientia. 37:1607-1618
- Publication Year :
- 2017
- Publisher :
- Elsevier BV, 2017.
-
Abstract
- This paper is concerned with the Diophantine properties of the sequence { ξ θ 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.
- Subjects :
- Sequence
Logarithm
General Mathematics
Diophantine equation
010102 general mathematics
General Physics and Astronomy
01 natural sciences
Combinatorics
Base (group theory)
symbols.namesake
Bernoulli's principle
Fourier transform
0103 physical sciences
symbols
010307 mathematical physics
0101 mathematics
Algebraic integer
Mathematics
Subjects
Details
- ISSN :
- 02529602
- Volume :
- 37
- Database :
- OpenAIRE
- Journal :
- Acta Mathematica Scientia
- Accession number :
- edsair.doi...........f22a5832bacd1ff83f9f4b8e9cad1136