Fourier Transform Of Self-similar Measures And Uniform Distribution Mod 1

The thesis is mainly concerned with the Fourier transforms of self-similar measures and uniform distribution mod 1 of certain sequences.A class of self-similar measures whose have uniform contractive ratios and satisfy certain arithmetic properties are considered.We focus on the decay rate of the Fourier transforms of these self-similar measures.As application,We obtain some results of the uniform distribution mod 1 for certain sequences in their supports.Let λ:= θ-1(θ>1)be the contractive ratio of the self-similar measure μλ,we investigate the decay rate of the Fourier transform of μλby studying the Diophantine properties of the sequence {ξθn}n≥1，where 1 ≤ ξ ＜θ To deal with the two cases where θ is a rational number or an algebraic integer in a unified manner,we establish a combinatorial lemma.Based on this,the decay rate of the Fourier transform of μλis proved to be logarithmic in both cases,which extends a previous work of Kershner and Bufetov-Solomyak on Bernoulli convolutions.Besides,we prove that the decay rate of the Fourier transform of the image measure ofμλunder a.C2 function(the second derivative larger than 0)is polynomial no mater what the arithmetic property of the contractive ratio is,which extends a result of Kaufman on Bernoulli convolutions and answers a question raised by Hochman and Shmerkin.As applications,we show a result similar to the Cassels-Schimdt theorem on normal numbers.An estimate on the discrepancy of certain sequences and an application in the set of uniqueness in trigonometrical series are also included.The thesis is organized as follows:In Chapter 1,some backgrounds on the Fourier transforms of self-similar measures and uniform distribution mod 1 on fractals are introduced.We also give a brief summary our work in this thesis.In Chapter 2,we present some basics needed in this thesis,including self-similar sets and self-similar measures,Fourier transforms of Bernoulli convolutions,as well as Pisot numbers and Salem numbers.Chapter 3 is devoted to the distribution of the sequence ｛ξθn｝n≥1。We established a combinatorial lemma，in particular,a.method on how to estimate the cardinality of the set {n:1≤n ≤ N,‖ξθn‖＞δ} is given.In the meanwhile,we indicate how to treat the two cases where θ is a rational number or an algebraic integer in a unified manner by using this combinatorial lemma.In Chapter 4,we point out the relationship between the distribution of the fraction-al part of the sequence ｛ξθn｝n≥1 and the Fourier transform of the relevant self-similar measure.By virtue of the combinatorial lemma established in the last chapter,we show that the decay rate of the Fourier transform of μλ is logarithmic when θ satisfies certain arithmetic properties;we also show that the decay rate of the Fourier transform of the phase measure of μλ under a C2 action is polynomial for arbitrary contractive ratio.Fi-nally,we give an application of these results in the set of uniqueness in trigonometrical series.Chapter 5 mainly focuses on uniform distribution mod 1 on fractals.By using Davenport-Erdos-Leveque Theorem,we prove that almost all points in the supports of the aforementioned self-similar measures are absolutely normal.Furthermore,we estimate the discrepancy of certain sequences by using Schmidt’s method.The last chapter contains some concluding remarks and open questions.