No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Fourier decay of self-similar measures and self-similar sets of uniqueness
... The situation in which H is cyclic has been considered by Brémont (see also the paper by Varjú and Yu [28]). We continue to consider the orientation preserving system Φ = {ϕ i (t) = r i t + a i } i=1 on R. Theorem 1.3 (Brémont, [5]). ...
... We point out that Solomyak [26] has recently shown that there exists E ⊂ (0, 1) of zero Hausdorff dimension so that when Φ is affinely irreducible and (r i ) i=1 / ∈ E, it holds that µ p has power decay for all p ∈ ∆. For explicit parameters, and under additional diophantine assumptions, Li and Sahlsten [19] and Varjú and Yu [28] have recently obtained logarithmic decay rate. In the present paper we are only interested in the complete characterization of self-similar IFSs generating non-Rajchman measures, and do not consider the Fourier rate of decay. ...
... Cleary ψ(G) = R in this case, and so ψ(G) = βZ for some β > 0. Let U ∈ O(d) be with (β, U ) ∈ G, and set A = 2 −β U . Under the additional technical assumption a 1 = 0, we show that condition (2) in the statement of Theorem 1.5 holds for the matrix A. The proof is a nontrivial extension of the argument used in [28] for the direction of Theorem 1.3 in which the IFS is assumed to generate a non-Rajchman measure. One of the main ingredients of that argument is a classical theorem of Pisot. ...
We establish a complete algebraic characterization of self-similar iterated function systems on , for which there exists a positive probability vector p so that the Fourier transform of the self-similar measure corresponding to and p does not tend to 0 at infinity.
... There is currently much interest in the theory of Fourier transforms of self-similar measures, see the definition in Section 4. Denote the reciprocal of the contraction ratio of self-similar measure by α, we know that the Fourier analytic properties of self-similar measures have close connections with the distribution of fractional part of the sequence (ξα n ) n≥1 , see for example [7,10,20]; for more recent results, see [1,19,23]. ...
... In this paper, we shall prove some results on the distribution of fractional parts of certain sequences and then give an application to Fourier decay of self-similar measures. In particular, we generalize a result of Varjú and Yu [23]. ...
Assume that is an algebraic number and is a real number. We are concerned with the distribution of the fractional parts of the sequence . Under various Diophantine conditions on and , we obtain lower bounds on the number n with for which the fractional part of the sequence fall into a prescribed region , extending several results in the literature. As an application, we show that the Fourier decay rate of some self-similar measures is logarithmic, generalizing a result of Varj\'{u} and Yu.
... See also [2], where this result is extended to self-conformal measures using a different method. The lacunary case was analyzed by Brémont [7], see also Varjú, Yu [55]. Finally, the problem was solved by Rapaport [40] for self-similar measures on R d . ...
... See [2] for a similar result under a different Diophantine condition. Polylogarithmic Fourier decay was also established by Varjú and Yu [55] for certain self-similar measures in the lacunary case. ...
... Various thermodynamic, renewal theoretic and additive combinatoric techniques have been built which enable a systematic study of Fourier transforms of fractal measures using e.g. the under nonlinearity of the system. Since [39,9], there has been a surge of activity in this topic in dynamics, metric number theory and fractal geometry to characterise measures with Fourier decay, such as for self-similar-and self-affine iterated function systems [36,37,61,11,67,54], self-conformal systems [55,2], hyperbolic dynamical systems [43,44,45,70], fractal measures arising from random processes such as Brownian motion and Liouville quantum gravity [26,27,24,57]. ...
... self-affine systems [37] and the recent work by Khalil [41] on exponential mixing of the geodesic flow on a geometrically finite locally symmetric space of negative curvature with respect to the Bowen-Margulis-Sullivan measure, where Fourier decay results are studied under non-concentration on hyperplanes. In the overlapping selfsimilar case, it is possible to obtain logarithmic Fourier decay [36,11,67], but the renewal theoretic method uses the Cauchy-Schwartz inequality in a way so that the non-concentration from purely derivatives is not strong enough to establish polynomial Fourier decay. ...
We prove a uniform spectral gap for complex transfer operators near the critical line associated to overlapping iterated function systems on the real line satisfying a Uniform Non-Integrability (UNI) condition. Our work extends that of Naud (2005) on spectral gaps for nonlinear Cantor sets to allow overlaps. The proof builds a new method to reduce the problem of the lack of Markov structure to average contraction of products of random Dolgopyat operators. This approach is inspired by a disintegration technique developed by Algom, the first author and Shmerkin in the study of normal numbers. As a consequence of the method of the second author and Stevens, our spectral gap result implies that the Fourier transform of any non-atomic self-conformal measure decays to zero at a polynomial rate for any iterated function system satisfying UNI. This latter result leads to Fractal Uncertainty Principles with arbitrary overlaps.
... See also [2], where this result is extended to self-conformal measures using a different method. The lacunary case was analysed by Brémont [7], see also Varjú, Yu [55]. Finally, the problem was solved by Rapaport [40] for self-similar measures on R d . ...
... See [2] for a similar result under a different Diophantine condition. Polylogarithmic Fourier decay was also established by Varjú and Yu [55] for certain self-similar measures in the lacunary case. ...
We discuss the problem of determining the dimension of self-similar sets and measures on . We focus on the developments of the last four years. At the end of the paper, we survey recent results about other aspects of self-similar measures including their Fourier decay and absolute continuity.
... A measure ν is said to have power Fourier decay if ν P D d . Many recent papers have been devoted to the question of Fourier decay for classes of "fractal" measures, see e.g., [2,9,11,12,13,18,23,3,1,25,17]. Here we continue this line of research, focusing on the class of homogeneous self-affine measures in R d . ...
... For "generic" choices of the probability vector p, assuming that D Ă Qpλq after an affine conjugation, this is also sufficient, but there are some exceptional cases of positive co-dimension. Varjú and Yu [25] proved logarithmic decay of the Fourier transform in the case when r j " λ n j for j ď m and 1{λ is algebraic, but not a Pisot or Salem number. In [23] we showed that outside a zero Hausdorff dimension exceptional set of parameters, all self-similar measures on R belong to D 1 ; however, the exceptional set is not explicit. ...
We show that for almost all d-tuples , with , any self-affine measure for a homogeneous non-degenerate iterated function system in , where is a diagonal matrix with the entries , has power Fourier decay at infinity.
... In the complementary case, when all contractions are powers of some r ∈ (0, 1), Brémont [5] proved that if these powers are coprime, then a non-atmoic self-similar measure can fail to be Rajchman only if r −1 is Pisot (in fact, Brémont also reproved the Li-Sahlsten Theorem, and carried out a refined analysis giving some cases when r −1 is Pisot yet the measure is Rajchman). Another proof of this fact was given by Varjú and Yu [42]. We remark that while Corollary 1.2 covers many IFS's as in Li and Sahlsten's paper (and can be adapted to cover all of them), it does not cover IFS's as in the works of Brémont and of Varjú and Yu, since these are always periodic. ...
... Li and Sahlsten [25] prove a logarithmic rate of decay assuming a certain Diophantine condition holds on a pair of independent contractions. A similar rate of decay is obtained in the work of Varjú and Yu [42] assuming the underlying contraction is not Pisot or Salem. Finally, in another very recent paper, by generalizing the Erdős-Kahane arguement, Solomyak [36] established polynomial Fourier decay for all non-atomic self-similar measures except for a zero Hausdorff dimensional exceptional set of contractions. ...
Let be a smooth IFS on an interval , where . We provide mild conditions on the derivative cocycle that ensure that all non-atomic self conformal measures are Rajchman measures, that is, their Fourier transform decays to 0 at infinity. This allows us to give many new examples of self conformal Rajchman measures, and also provides a unified proof to several pre-existing results. For example, we show that if is and admits a non-atomic non-Rajchman self conformal measure, then it is conjugate to a self similar IFS satisfying: This is closely related to the work of Bourgain-Dyatlov. We also prove that if is self similar and does have not this form, then any smooth image of a non atomic self similar measure is Rajchman, assuming the derivative never vanishes. This complements a classical Theorem of Kaufman about homogeneous IFS's, and extends in many cases recent results of Li-Sahlsten about the Rajchman property in the presence of independent . The proof relies on a version of the local limit Theorem for the derivative cocycle, that is adapted from the work of Benoist-Quint.
... The intermediate case, namely, fractal measures is a very difficult problem and relates to classifying the sets of uniqueness and multiplicity for trigonometric series [8,25,30], Diophantine approximation [40], and Fourier dimension [35,16]. For the middle-third Cantor measure, the Fourier transform cannot decay at infinity due to invariance under ×3 mod 1, but some other fractal measures such as random measures (see Salem's work [42] on random Cantor measures or Kahane's work on Brownian motion [20,21]), measures arising in Diophantine approximation (see Kaufman's measures on well and badly approximable numbers [23,22]), non-Pisot Bernoulli convolutions [42], and self-similar measures with suitable irrationality properties [30,7,48] all exhibit decay of Fourier coefficients. ...
... Establishing rates for these renewal theorems, at least in higher dimensions, one needs to have the similar sum-product bounds available as we have. Further characterisations to these were done recently by Brémont [7] and Varjú-Yu [48]. Moreover, using the Erdös-Kahane method Solomyak [47] proved polynomial Fourier decay for all non-atomic self-similar measures except for a zero-Hausdorff dimensional exceptional set of parameters. ...
FOR THE LATEST FILE VERSION PLEASE SEE ARXIV.
https://arxiv.org/abs/2009.01703
We study the Fourier transforms of Gibbs measures for uniformly expanding maps T of bounded distortions on Cantor sets with strong separation condition. When T is totally non-linear and Hausdorff dimension of is large enough, then decays at a polynomial rate as .
Edit: we believe that the dimension assumption can be removed.
Version 2: corrections made to main assumptions. Ammended some typos, and expanded the introduciton.
... ) We end this introduction with a few remarks about the implications Theorem 2 has on the Fourier decay problem for fractal measures. Recall that a probability measure ν on R is called a Rajchman measure if lim |q|→∞ ν(q) = 0. Combining recent breakthroughs on the Fourier decay problem [13,25,20,4,3,1,5] we know that if a C ω (R) self-conformal IFS admits a non-Rajchman self-conformal measure then: The IFS must be self-similar; It must have contraction ratios that are all powers of some r ∈ (0, 1) such that r −1 is a Pisot number; And, up to affine conjugation, all translates are in Q(r ). Now, it is natural to ask about analogues of this result in C r (R)-regularity, 1 ≤ r ≤ ∞ -see e.g. ...
Motivated by a question of M. Hochman, we construct examples of hyperbolic IFSs on [0,1] where linear and non-linear behaviour coexist. Namely, for every we exhibit the existence of a -smooth IFS such that on the attractor and for every , yet is not -smooth for any , nor -conjugate to self-similar. We provide a complete classification of these systems. Furthermore, when , we give a necessary and sufficient Livsic-like matching condition for a self-conformal -smooth IFS to be conjugated to one of these systems having on the attractor, for every . We also show that this condition fails to ensure the existence of a -conjugacy in mere -regularity.
... We do note that [4,10] say nothing about selfsimilar measures, which is an important case. For recent progress on decay for self-similar measures we refer to [54,49,61,5,44,17,19,18,59] and references therein. Theorem 1.1 also has counterparts in higher dimensions. ...
Let be a self-conformal IFS on the plane, satisfying some mild non-linearity and irreducibility conditions. We prove a uniform spectral gap estimate for the transfer operator corresponding to the derivative cocycle and every given self-conformal measure. Building on this result, we establish polynomial Fourier decay for any such measure. Our technique is based on a refinement of a method of Oh-Winter (2017) where we do not require separation from the IFS or the Federer property for the underlying measure.
... We conclude this part with a recent theorem of Varjú and Yu [13] which plays a major role in our proof that T b -invariant Rajchman measure need not be Parry. THEOREM 7.1. ...
We prove measure rigidity for multiplicatively independent pairs when and is a ‘specified’ real number (the b -expansion of 1 has a tail or bounded runs of 0 s) under a positive entropy condition. This is done by proving a mean decay of the Fourier series of the point masses average along orbits. We also prove a quantitative version of this decay under stronger conditions on the invariant measure. The quantitative version together with the invariance of the limit measure is a step toward a general Host-type pointwise equidistribution theorem in which the equidistribution is for Parry measure instead of Lebesgue. We show that finite memory length measures on the a -shift meet the mentioned conditions for mean convergence. Our main proof relies on techniques of Hochman.
... In [19], Solomyak proved that almost every self-similar measures on the real line has a power decay of the Fourier transform at infinity. Varjú and Yu [24] provided quantitative decay rates of Fourier transform of some self-similar measures, using random walks on lattices and Diophantine approximation in number fields. Considering the interplay between the behavior of Fourier transform and the absolute continuity of a measure, that the faster the Fourier transform of a measure tends to zero the more regular the measure is, we focus our consideration on the Fourier dimensions of in-homogeneous self-similar measures. ...
The in-homogeneous self-similar measure is defined by the relation where is a probability vector, each , , is a contraction similarity, and is a Borel probability measure on with compact support. In this paper, we study the asymptotic behavior of the Fourier transforms of in-homogeneous self-similar measures. We obtain non-trivial lower and upper bounds for the qth lower Fourier dimensions of the in-homogeneous self-similar measures without any separation conditions. Moreover, if the IFS satisfies some separation conditions, the lower bounds for the qth lower Fourier dimensions can be improved. These results confirm conjecture 2.5 and give a positive answer to the question 2.7 in Olsen and Snigireva’s paper (Math Proc Camb Philos Soc 144(2):465–493, 2008).
... Returning to the study of sets with positive Fourier dimension and of sets of unicity, let us also mention that a there has been a lot of interest for Cantor sets appearing from linear IFS. Related work includes [LS19a], [LS19b], [So19], [Br19], [VY20], and [Ra21]. ...
Let be a (convex-)cocompact group of isometries of the hyperbolic space , let be the associated hyperbolic manifold, and consider a real valued potential F on its unit tangent bundle . Under a natural regularity condition on F, we prove that the associated -Patterson-Sullivan densities are stationary measures with exponential moment for some random walk on . As a consequence, when M is a surface, the associated equilibrium state for the geodesic flow on exhibit "Fourier decay", in the sense that a large class of oscillatory integrals involving it satisfies power decay. It follows that the non-wandering set of the geodesic flow on convex-cocompact hyperbolic surfaces has positive Fourier dimension, in a sense made precise in the appendix.
... Furthermore, Φ is affinely conjugated to an IFS that has all of its translations in Q(r). For more recent results on Fourier decay for self similar measures we refer to [45,40,50,4,37,16,18,17,49] and references therein. ...
We show that every self conformal measure with respect to a IFS has polynomial Fourier decay under some mild and natural non-linearity conditions. In particular, every such measure has polynomial decay if is and contains a non-affine map. A key ingredient in our argument is a cocycle version of Dolgopyat's method, that does not require the cylinder covering of the attractor to be a Markov partition. It is used to obtain spectral gap-type estimates for the transfer operator, which in turn imply a renewal theorem with an exponential error term in the spirit of Li (2022).
... We conclude this part with a recent theorem of Varjú and Yu [VY20] which plays a major role in our proof that T b -invariant Rajchman measure need not be Parry, Theorem 7.1. Let k ≥ 2 be an integer. ...
We prove measure rigidity for multiplicatively independent pairs when and is a ``specified'' real number (the b-expansion of 1 has a tail or bounded runs of 0's) under a positive entropy condition. This is done by proving a mean decay of the Fourier series of the point masses average along orbits. We also prove a quantitative version of this decay under stronger conditions on the invariant measure. The quantitative version together with the invariance of the limit measure is a step toward a general Host-type pointwise equidistribution theorem in which the equidistribution is for Parry measure instead of Lebesgue. We show that finite memory length measures on the a-shift meet the mentioned conditions for mean convergence. Our main proof relies on techniques of Hochman.
... We are also grateful to P. Varjú for pointing out an error in a first version of Theorem 2.7. His work with H. Yu [VY20], completes Section 6.2 and fully characterizes sets of uniqueness among self-similar sets. ...
... Using transfer operators, Jordan and Sahlsten [JS16] studied invariant measures for the Gauss map. Li [Li17] introduced a method based on renewal theorems for random walks to study stationary measures, that leads to several results in the linear IFS case for self-affine measures, see [LS19] and related work [So19,Br19,VY20,Ra21]. See also [ARW20] for a study of the nonlinear IFS case in dimension one. ...
Let be a hyperbolic rational map of degree , and let be its Julia set. We prove that J always has positive Fourier dimension. The case where J is included in a circle follows from a recent work of Sahlsten and Stevens (Fourier transform and expanding maps on Cantor sets preprint. arXiv:2009.01703, 2020). In the case where J is not included in a circle, we prove that a large family of probability measures supported on J exhibit polynomial Fourier decay: our result applies in particular to the measure of maximal entropy and to the conformal measure.
... For "generic" choices of the probability vector p, assuming that D Q. / after an affine conjugation, this is also sufficient, but there are some exceptional cases of positive co-dimension. Varjú and Yu [25] proved logarithmic decay of the Fourier transform in the case when r j D n j for j Ä m and 1= is algebraic, but not a Pisot or Salem number. In [23], we showed that outside a zero Hausdorff dimension exceptional set of parameters, all self-similar measures on R belong to D 1 ; however, the exceptional set is not explicit. ...
We show that for Lebesgue almost all d -tuples (\theta_1,\ldots,\theta_d) , with |\theta_j|>1 , any self-affine measure for a homogeneous non-degenerate iterated function system \{Ax+a_j\}_{j=1}^m in \mathbb{R}^d , where A^{-1} is a diagonal matrix with the entries (\theta_1,\ldots,\theta_d) , has power Fourier decay at infinity.
... Based on this theorem, many such results [6, 7, 11-13, 20, 29] have been proved for Riesz products or measures with an infinite convolution structure. We also refer to [2,10,23,25,26,31,32,34,36] for some recent works on the Fourier decay for fractal measures. ...
In this paper, we study the pointwise equidistribution properties of measures μ p defined by digit restrictions on the b -adic expansion, where b ⩾ 2 is an integer. We prove that, if a sequence ( α n ) n ⩾ 1 satisfies a certain b -adic diversity condition, then the sequence ( α n x ) n ⩾ 1 is uniformly distributed modulo one for μ p -a.e. x . We also find some sufficient conditions to ensure the b -adic diversity. Moreover, we apply these results to establish the b -adic diversity for the sequences that can be written as certain combination of polynomial and exponential functions.
... In [19] it was shown that for any homogeneous IFS with non-atomic self-similar measure µ, if g is a C 2 function satisfying g ′′ > 0 then the Fourier transform of gµ decays to zero sufficiently quickly so that the Davenport-Erdős-LeVeque theorem applies. For dynamically defined measures there are many recent results that establish a sufficiently fast rate of decay for this result to apply: [1,13,27,16,30,31,24] to name just a few examples (see e.g. [2] for many more references). ...
Let be a self-similar IFS on and let be a Pisot number. We prove that if for some i then for every diffeomorphism g and every non-atomic self similar measure , the measure is supported on numbers that are normal in base .
... This includes the work of Erdős [15] and Kahane [20] about polynomial decay being typical for Bernoulli convolutions (the Erdős-Kahane argument) and the more recent works [11,8,12] about rates in some explicit examples of Bernoulli convolutions. In the self-similar case when all contractions are powers of some r ∈ (0, 1), Varjú-Yu [37] proved logarithmic decay as long as r −1 is not a Pisot or a Salem number. We also mention the work of Kaufman [21] and Mosquera-Shmerkin [29] about polynomial Fourier decay for C 2 IFS's that arise by conjugating homogeneous (that are never Diophantine) self-similar IFS's. ...
We prove that the Fourier transform of a self conformal measure on decays to 0 at infinity at a logarithmic rate, unless the following holds: The underlying IFS is smoothly conjugated to an IFS that both acts linearly on its attractor and contracts by scales that are simultaneously Liouvillian. Our key technical result is an effective version of a local limit Theorem for cocycles with moderate deviations due to Benoist-Quint (2016), that is of independent interest.
... The rate at which the Fourier transform of a fractal measure decays to zero is a well studied problem. For more on this topic we refer the reader to the papers [12,14,16,18,20,22,26] and the references therein. ...
In this paper we prove that if is an equicontractive iterated function system and b is a positive integer satisfying then almost every x is normal in base b for any non-atomic self-similar measure of .
... In general, when the linear parts are not the same, one can still write down the Fourier coefficient as the expectation of a random walk, see for example [28]. ...
In this paper, we consider a problem of counting rational points near self-similar sets. Let be an integer. We shall show that for some self-similar measures on , the set of rational points is 'equidistributed' in a sense that will be introduced in this paper. This implies that an inhomogeneous Khinchine convergence type result can be proved for those measures. In particular, for n=1 and large enough integers p, the above holds for the middle-pth Cantor measure, i.e. the natural Hausdorff measure on the set of numbers whose base p expansions do not have digit Furthermore, we partially proved a conjecture of Bugeaud and Durand for the middle-pth Cantor set and this also answers a question posed by Levesley, Salp and Velani. Our method includes a fine analysis of the Fourier coefficients of self-similar measures together with an Erd\H{o}s-Kahane type argument. We will also provide a numerical argument to show that is sufficient for the above conclusions. In fact, is already enough for most of the above conclusions.
We solve a long-standing open problem of determining the Fourier dimension of the Mandelbrot canonical cascade measure (MCCM). This problem of significant interest was raised by Mandelbrot in 1976 and reiterated by Kahane in 1993. Specifically, we derive the exact formula for the Fourier dimension of the MCCM for random weights W satisfying the condition for all . As a corollary, we prove that the MCCM is Salem if and only if the random weight has a specific two-point distribution. In addition, we show that the MCCM is Rajchman with polynomial Fourier decay whenever the random weight satisfies for some . As a consequence, we discover that, in the Biggins-Kyprianou's boundary case, the Fourier dimension of the MCCM exhibits a second order phase transition at the inverse temperature ; we establish the upper Frostman regularity for MCCM; and we obtain a Fourier restriction estimate for MCCM. While the precise Fourier dimension formula seems to be a little magical in the realm of cascade theory, it turns out to be fairly natural in the context of vector-valued martingales. Indeed, the vector-valued martingales will play key roles in our new formalism of actions by multiplicative cascades on finitely additive vector measures, which is the major novelty of this paper.
We study the law of random self-similar series defined above an irrational rotation on the Circle. This provides a natural class of continuous singular non-Rajchman measures.
In this article we study the generalized Fourier dimension of the set of Liouville numbers . Being a set of zero Hausdorff dimension, the analysis has to be done at the level of functions with a slow decay at infinity acting as control for the Fourier transform of (Rajchman) measures supported on . We give an almost complete characterization of admissible decays for this set in terms of comparison to power-like functions. This work can be seen as the ``Fourier side'' of the analysis made by Olsen and Renfro regarding the generalized Hausdorff dimension using gauge functions. We also provide an approach to deal with the problem of classifying oscillating candidates for a Fourier decay for relying on its translation invariance property.
A real number x is considered normal in an integer base if its digit expansion in this base is “equitable”, ensuring that for each , every ordered sequence of k digits from occurs in the digit expansion of x with the same limiting frequency. Borel’s classical result [4] asserts that Lebesgue-almost every is normal in every base . This paper serves as a case study of the measure-theoretic properties of Lebesgue-null sets containing numbers that are normal only in certain bases. We consider the set of reals that are normal in odd bases but not in even ones. This set has full Hausdorff dimension [30] but zero Fourier dimension. The latter condition means that cannot support a probability measure whose Fourier transform has power decay at infinity. Our main result is that supports a Rajchman measure , whose Fourier transform approaches 0 as by definiton, albeit slower than any negative power of . Moreover, the decay rate of is essentially optimal, subject to the constraints of its support. The methods draw inspiration from the number-theoretic results of Schmidt [38] and a construction of Lyons [24]. As a consequence, emerges as a set of multiplicity, in the sense of Fourier analysis. This addresses a question posed by Kahane and Salem [17] in the special case of .
We establish a complete algebraic characterization of self-similar iterated function systems Φ on Rd, for which there exists a positive probability vector p so that the Fourier transform of the self-similar measure corresponding to Φ and p does not tend to 0 at infinity.
Let {fi(x)=si⋅x+ti} be a self-similar IFS on R and let β>1 be a Pisot number. We prove that if log|si|logβ∉Q for some i then for every C1 diffeomorphism g and every non-atomic self-similar measure μ, the measure gμ is supported on numbers that are normal in base β.
Let Φ be a C1+γ smooth IFS on R, where γ>0. We provide mild conditions on the derivative cocycle that ensure that every self conformal measure is supported on points x that are absolutely normal. That is, for every integer p≥2 the sequence {pkx}k∈N equidistributes modulo 1. We thus extend several state of the art results of Hochman and Shmerkin [29] about the prevalence of normal numbers in fractals. When Φ is self-similar we show that the set of absolutely normal numbers has full Hausdorff dimension in its attractor, unless Φ has an explicit structure that is associated with some integer n≥2. These conditions on the derivative cocycle are also shown to imply that every self conformal measure is a Rajchman measure, that is, its Fourier transform decays to 0 at infinity. When Φ is self similar and satisfies a certain Diophantine condition, we establish a logarithmic rate of decay.
This book presents state-of-the-art research on the distribution modulo one of sequences of integral powers of real numbers and related topics. Most of the results have never before appeared in one book and many of them were proved only during the last decade. Topics covered include the distribution modulo one of the integral powers of 3/2 and the frequency of occurrence of each digit in the decimal expansion of the square root of two. The author takes a point of view from combinatorics on words and introduces a variety of techniques, including explicit constructions of normal numbers, Schmidt's games, Riesz product measures and transcendence results. With numerous exercises, the book is ideal for graduate courses on Diophantine approximation or as an introduction to distribution modulo one for non-experts. Specialists will appreciate the inclusion of over 50 open problems and the rich and comprehensive bibliography of over 700 references.
This unified account of various aspects of a powerful classical method, easy to understand in its simplest forms, is illustrated by applications in several areas of number theory. As well as including diophantine approximation and transcendence, which were mainly responsible for its invention, the author places the method in a broader context by exploring its application in other areas, such as exponential sums and counting problems in both finite fields and the field of rationals. Throughout the book, the method is explained in a 'molecular' fashion, where key ideas are introduced independently. Each application is the most elementary significant example of its kind and appears with detailed references to subsequent developments, making it accessible to advanced undergraduates as well as postgraduate students in number theory or related areas. It provides over 700 exercises both guiding and challenging, while the broad array of applications should interest professionals in fields from number theory to algebraic geometry.
Let {F} be a self-similar set on \mathbb{R} associated to contractions {f_j(x) = r_j x + b_j} , {j \in \mathcal{A}} , for some finite \mathcal{A} , such that {F} is not a singleton. We prove that if {\log r_i / {\log r_j}} is irrational for some {i \neq j} , then {F} is a set of multiplicity, that is, trigonometric series are not in general unique in the complement of {F} . No separation conditions are assumed on {F} . We establish our result by showing that every self-similar measure {\mu} on {F} is a Rajchman measure: the Fourier transform {\widehat{\mu}(\xi) \to 0} as {|\xi| \to \infty} . The rate of {\widehat{\mu}(\xi) \to 0} is also shown to be logarithmic if {\log r_i / {\log r_j}} is diophantine for some {i \neq j} . The proof is based on quantitative renewal theorems for stopping times of random walks on \mathbb{R} .
The study of sets of uniqueness for trigonometric series has a long history, originating in the work of Riemann, Heine, and Cantor in the mid-nineteenth century. Since then it has been a fertile ground for numerous investigations involving real analysis, classical and abstract harmonic analysis, measure theory, functional analysis and number theory. In this book are developed the intriguing and surprising connections that the subject has with descriptive set theory. These have only been discovered recently and the authors present here this novel theory which leads to many new results concerning the structure of sets of uniqueness and include solutions to some of the classical problems in this area. In order to make the material accessible to logicians, set theorists and analysts, the authors have covered in some detail large parts of the classical and modern theory of sets of uniqueness as well as the relevant parts of descriptive set theory. Thus the book is essentially self-contained and will make an excellent introduction to the subject for graduate students and research workers in set theory and analysis.
the attention of The publication of Charles Pisot's thesis in 1938 brought to the mathematical community those marvelous numbers now known as the Pisot numbers (or the Pisot-Vijayaraghavan numbers). Although these numbers had been discovered earlier by A. Thue and then by G. H. Hardy, it was Pisot's result in that paper of 1938 that provided the link to harmonic analysis, as discovered by Raphael Salem and described in a series of papers in the 1940s. In one of these papers, Salem introduced the related class of numbers, now universally known as the Salem numbers. These two sets of algebraic numbers are distinguished by some striking arith metic properties that account for their appearance in many diverse areas of mathematics: harmonic analysis, ergodic theory, dynamical systems and alge braic groups. Until now, the best known and most accessible introduction to these num bers has been the beautiful little monograph of Salem, Algebraic Numbers and Fourier Analysis, first published in 1963. Since the publication of Salem's book, however, there has been much progress in the study of these numbers. Pisot had long expressed the desire to publish an up-to-date account of this work, but his death in 1984 left this task unfulfilled.
This paper is concerned with the Diophantine properties of the sequence{ξθⁿ}, where1 ≤ 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 measuresμλ withλ = 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 thatμλ 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.
We owe to MINKOWSKI the fertile observation that certain results which can be made almost intuitive by the consideration of figures in n-dimensional euclidean space have far-reaching consequences in diverse branches of number theory. For example, he simplified the theory of units in algebraic number fields and both simplified and extended the theory of the approximation of irrational numbers by rational ones (Diophantine Approximation). This new branch of number theory, which MINKOWSKI christened “The Geometry of Numbers”, has developed into an independent branch of number-theory which, indeed, has many applications elsewhere but which is well worth studying for its own sake.
The paper gives first quantitative estimates on the modulus of continuity of
the spectral measure for weakly mixing suspension flows over substitution
automorphisms. The main results are, first, a Hoelder estimate for the spectral
measure of almost all suspension flows with a piecewise constant roof function;
second, a log-Hoelder estimate for self-similar suspension flows; and, third, a
Hoelder asymptotic expansion of the spectral measure at zero for such flows.
The second result implies log-Hoelder estimates for the spectral measures of
translation flows along stable foliations of pseudo-Anosov automorphisms. The
Appendix explains the connection of these results with the theory of Bernoulli
convolutions.