DECAY RATE OF FOURIER TRANSFORMS OF SOME SELF-SIMILAR MEASURES

doi:10.1016/S0252-9602(17)30094-2

Abstract

Abstract:

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 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.

Key words: self-similar measures, Fourier transforms, decay rate, normal numbers

Xiang GAO, Jihua MA. DECAY RATE OF FOURIER TRANSFORMS OF SOME SELF-SIMILAR MEASURES[J].Acta mathematica scientia,Series B, 2017, 37(6): 1607-1618.

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References

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