Digit Set Research Articles

Spectrality and non-spectrality of a class of Moran measures

Let {Mn}n=1∞∈M2(R) be a sequence of real expanding matrices and {Dn}n=1∞ be a sequence of digit sets withDn={(00),(anbn),(cndn)}, where |andn−bncn|=ϕ(n)∈Z, and letμ{Mn},{Dn}=δM1−1D1⁎δ(M1M2)−1D2⁎δ(M1M2M3)−1D3⁎⋯ be the associated Moran measure. This paper establishes a necessary and sufficient condition for μMn,Dn to be a spectral measure when each Mn is a diagonal matrix, and characterizes the structure of the spectrum Λ. Furthermore, we demonstrate that for any sequence of matrices {Mn}n=1∞ ∈ M2(Z) with det⁡(Mn)∉3Z, the space L2(μ{Mn},{Dn}) contains at most 9 mutually orthogonal exponential functions. The precise maximum cardinality of such functions is also determined.

Spectral structure of a class of self-similar spectral measures with product form digit sets

Spectral structure of a class of self-similar spectral measures with product form digit sets

On matching and periodicity for (N,α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(N,\\alpha )$$\\end{document}-expansions

Recently a new class of continued fraction algorithms, the (N,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(N,\\alpha $$\\end{document})-expansions, was introduced in Kraaikamp and Langeveld (J Math Anal Appl 454(1):106–126, 2017) for each N∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\in \\mathbb {N}$$\\end{document}, N≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 2$$\\end{document} and α∈(0,N-1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (0,\\sqrt{N}-1]$$\\end{document}. Each of these continued fraction algorithms has only finitely many possible digits. These (N,α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(N,\\alpha )$$\\end{document}-expansions ‘behave’ very different from many other (classical) continued fraction algorithms; see also Chen and Kraaikamp (Matching of orbits of certain n-expansions with a finite set of digits (2022). To appear in Tohoku Math. J arXiv:2209.08882), de Jonge and Kraaikamp (Integers 23:17, 2023), de Jonge et al. (Monatsh Math 198(1):79–119, 2022), Nakada (Tokyo J Math 4(2):399–426, 1981) for examples and results. In this paper we will show that when all digits in the digit set are co-prime with N, which occurs in specified intervals of the parameter space, something extraordinary happens. Rational numbers and certain quadratic irrationals will not have a periodic expansion. Furthermore, there are no matching intervals in these regions. This contrasts sharply with the regular continued fraction and more classical parameterised continued fraction algorithms, for which often matching is shown to hold for almost every parameter. On the other hand, for α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} small enough, all rationals have an eventually periodic expansion with period 1. This happens for all α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} when N=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=2$$\\end{document}. We also find infinitely many matching intervals for N=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=2$$\\end{document}, as well as rationals that are not contained in any matching interval.

Spectrality of homogeneous Moran measures on the plane

LetM=ρ1−1a0ρ2−1 be a real upper triangular expanding matrix and D=00,d10,d2d3 be a three-element real digit set with d1d3≠0 , and let {nk}k=1∞ be a sequence of positive integers with upper bound. The infinite convolutions μM,D,{nk}=δM−n1D∗δM−(n1+n2)D∗⋯∗δM−(n1+⋯+nk)D∗⋯ converges weakly to a Borel probability measure (homogeneous Moran measure). In this paper, we study the existence of exponential orthonormal basis for L2(μM,D,{nk}). A necessary and sufficient condition for μM,D,{nk} to be a spectral measure is established.

Scaling spectrum of a class of self-similar measures with product form on ℝ

Abstract Let p, q, N ≥ 2 {N\geq 2} be three positive integers and let D = { 0 , 1 , … , N - 1 } ⊕ N p ⁢ { 0 , 1 , … , N - 1 } {D=\{0,1,\ldots,N-1\}\oplus N^{p}\{0,1,\ldots,N-1\}} be a product form digit set. It is well known that if q ∤ p {q\nmid p} , then the self-similar measure μ N q , D {\mu_{N^{q},D}} generated by the iterated function system { ( N q ) - 1 ⁢ ( x + d ) } d ∈ D , x ∈ ℝ {\{(N^{q})^{-1}(x+d)\}_{d\in D,x\in\mathbb{R}}} is a spectral measure with a spectrum Λ ⁢ ( N q , C ) = { ∑ i = 0 finite c i ⁢ N q ⁢ i : c i ∈ C } , \Lambda(N^{q},C)=\Bigg{\{}\sum_{i=0}^{\text{finite}}c_{i}N^{qi}:c_{i}\in C% \Bigg{\}}, where C = N q - p - 1 ⁢ D {C=N^{q-p-1}D} . In this paper, based on the properties of cyclic groups in number theory, we give some conditions on real number t under which the scaling set t ⁢ Λ ⁢ ( N q , C ) {t\Lambda(N^{q},C)} is also a spectrum of μ N q , D {\mu_{N^{q},D}} .

A class of self-affine tiles in [formula omitted] that are tame balls revisited

The author of this paper and coauthors in 2022 studied a family of self-affine tiles in Rd with noncollinear digit sets, and gave a sufficient and necessary condition for such tiles to be tame balls. We in this paper mainly present a simpler proof of such equivalent condition. We replace quadric surfaces by some zigzag planes, and redefine the quasi-invariant plane which plays a key role in the construction of the desired homeomorphism. This adjustment greatly simplifies the proof.

Non-expansive matrix number systems with bases similar to certain Jordan blocks

We study representations of integral vectors in a number system with a matrix base M and vector digits. We focus on the case when M is equal or similar to Jn, the Jordan block with eigenvalue 1 and dimension n. If M=J2, we classify all digit sets of size two allowing representation for all of Z2. For M=Jn with n≥3, we show that a digit set of size three suffice to represent all of Zn. For bases M similar to Jn, n≥2, we construct a digit set of size n such that all of Zn is represented. The language of words representing the zero vector with M=J2 and the digits (0,±1)T is shown not to be context-free, but to be recognizable by a Turing machine with logarithmic memory.

Some Results on Planar Self-Affine Measures with Collinear Digit Sets

Some Results on Planar Self-Affine Measures with Collinear Digit Sets

Two-Stage Feature Generator for Handwritten Digit Classification.

In this paper, a novel feature generator framework is proposed for handwritten digit classification. The proposed framework includes a two-stage cascaded feature generator. The first stage is based on principal component analysis (PCA), which generates projected data on principal components as features. The second one is constructed by a partially trained neural network (PTNN), which uses projected data as inputs and generates hidden layer outputs as features. The features obtained from the PCA and PTNN-based feature generator are tested on the MNIST and USPS datasets designed for handwritten digit sets. Minimum distance classifier (MDC) and support vector machine (SVM) methods are exploited as classifiers for the obtained features in association with this framework. The performance evaluation results show that the proposed framework outperforms the state-of-the-art techniques and achieves accuracies of 99.9815% and 99.9863% on the MNIST and USPS datasets, respectively. The results also show that the proposed framework achieves almost perfect accuracies, even with significantly small training data sizes.

Infinite orthogonal exponentials for a class of Moran measures

For a sequence [Formula: see text] of expanding matrices with [Formula: see text] and a sequence [Formula: see text] of finite digit sets with [Formula: see text], the Moran measure [Formula: see text] is defined by the infinite convolution [Formula: see text] and the convergence is in the weak sense. Under some additional assumptions, we show that [Formula: see text] contains an infinite orthogonal set of exponential functions if and only if there exists an infinite subsequence [Formula: see text] of [Formula: see text] such that [Formula: see text] for any [Formula: see text], where [Formula: see text]. This extends the results of [J. L. Li, A necessary and sufficient condition for the finite [Formula: see text]-orthogonality, Sci. China Math. 58 (2015) 2541–2548].

Dynamics of Ostrowski skew-product: 1. Limit laws and Hausdorff dimensions

We present a dynamical study of Ostrowski’s map based on the use of transfer operators. The Ostrowski dynamical system is obtained as a skew-product of the Gauss map (it has the Gauss map as a base and intervals as fibers) and produces expansions of real numbers with respect to an irrational base given by continued fractions. By studying spectral properties of the associated transfer operators, we show that the absolutely continuous invariant measure of the Ostrowski dynamical system has exponential mixing properties. We deduce a central limit theorem for random variables of an arithmetic nature, and motivated by applications in inhomogeneous Diophantine approximation, we also get Bowen–Ruelle type implicit estimates in terms of spectral elements for the Hausdorff dimension of a bounded digit set.

Beurling densities of regular maximal orthogonal sets of self-similar spectral measure with consecutive digit sets

Abstract Beurling density plays a key role in the study of frame-spectrality of normalized Lebesgue measure restricted to a set. Accordingly, in this paper, the authors study the s-Beurling densities of regular maximal orthogonal sets of a class of self-similar spectral measures, where s is the Hausdorff dimension of its support and obtain their exact upper bound of the densities.

Product-form Hadamard triples and its spectral self-similar measures

In a previous work by Łaba and Wang, it was proved that whenever there is a Hadamard triple (N,D,L), then the associated one-dimensional self-similar measure μN,D generated by maps N−1(x+d) with d∈D, is a spectral measure. In this paper, we introduce product-form digit sets for finitely many Hadamard triples (N,Ak,Lk) by putting each triple into different scales of N. Our main result is to prove that the associated self-similar measure μN,D is a spectral measure. This result allows us to show that product-form self-similar tiles are spectral sets as long as the tiles in the group ZN obey the Coven-Meyerowitz (T1), (T2) tiling condition. Moreover, we show that all self-similar tiles with N=pαq are spectral sets, answering a question by Fu, He and Lau in 2015. Finally, our results allow us to offer new singular spectral measures not generated by a single Hadamard triple. Such new examples allow us to classify all spectral self-similar measures generated by four equi-contraction maps, which will appear in a forthcoming paper.

Spectrality of Cantor–Moran measures with three-element digit sets

Abstract Let 0 < ρ < 1 {0<\rho<1} and let { a j , b j , n j } j = 1 ∞ {\{a_{j},b_{j},n_{j}\}_{j=1}^{\infty}} be a sequence of positive integers with an upper bound. Associated with them, there exists a unique Borel probability measure μ ρ , { 0 , a j , b j } , { n j } {\mu_{\rho,\{0,a_{j},b_{j}\},\{n_{j}\}}} generated by the following infinite convolution of discrete measures: μ ρ , { 0 , a j , b j } , { n j } = δ ρ n 1 ⁢ { 0 , a 1 , b 1 } ∗ δ ρ n 1 + n 2 ⁢ { 0 , a 2 , b 2 } ∗ δ ρ n 1 + n 2 + n 3 ⁢ { 0 , a 3 , b 3 } ∗ ⋯ , \mu_{\rho,\{0,a_{j},b_{j}\},\{n_{j}\}}=\delta_{\rho^{n_{1}}\{0,a_{1},b_{1}\}}% \ast\delta_{\rho^{n_{1}+n_{2}}\{0,a_{2},b_{2}\}}\ast\delta_{\rho^{n_{1}+n_{2}+% n_{3}}\{0,a_{3},b_{3}\}}\ast\cdots, where gcd ⁡ ( a j , b j ) = 1 {\gcd(a_{j},b_{j})=1} for all j ∈ ℕ {j\in{\mathbb{N}}} . In this paper, we show that L 2 ⁢ ( μ ρ , { 0 , a j , b j } , { n j } ) {L^{2}(\mu_{\rho,\{0,a_{j},b_{j}\},\{n_{j}\}})} admits an exponential orthonormal basis if and only if the following two conditions are satisfied: (i) { a j , b j } ≡ { ± 1 } ⁢ ( mod ⁢ 3 ) {\{a_{j},b_{j}\}\equiv\{\pm 1\}~{}(\mathrm{mod}~{}3)} for all j ≥ 1 {j\geq 1} ; (ii) there exists a natural number r such that ρ - r ∈ 3 ⁢ ℕ {\rho^{-r}\in 3{\mathbb{N}}} and n j ∈ r ⁢ ℕ {n_{j}\in r{\mathbb{N}}} for all j ≥ 2 {j\geq 2} .

Complete Characterization of Polyhedral Self-Affine Tiles

A self-affine tile is a compact set Gsubset {mathbb R}^d that admits a partition (tiling) by parallel shifts of the set M^{-1}G, where M is an expanding matrix. We find all self-affine tiles which are polyhedral sets, i.e., unions of finitely many convex polyhedra. It is shown that there exists an infinite family of such polyhedral sets, not affinely equivalent to each other. A special attention is paid to integral self-affine tiles with standard digit sets, when the matrix M and the translation vectors are integer. Applications to the approximation theory and to the functional analysis are discussed.

From positional representation of numbers to positional representation of vectors

To represent real m-dimensional vectors, a positional vector system given by a non-singular matrix M ∈ ℤm×m and a digit set Ɗ ⊂ ℤm is used. If m = 1, the system coincides with the well known numeration system used to represent real numbers. We study some properties of the vector systems which are transformable from the case m = 1 to higher dimensions. We focus on an algorithm for parallel addition and on systems allowing an eventually periodic representation of vectors with rational coordinates.

On zeros and spectral property of self-affine measures

In this paper, we study the spectral property of self-affine measures generated by an expanding integer matrix and a finite digit set , i.e. investigate whether the function in has a Fourier expansion. Let and for some integer . Through the discussion of the relationships between and , we establish some criteria to determine whether is spectral or non-spectral. Indeed, under those suitable assumptions, if is a spectral measure, we find its spectrum; otherwise, we give the maximum number of orthogonal exponential functions in . As an application, our results can contain some well-known conclusions.

Constructing bi-Lipschitz maps between Bedford-McMullen carpets via symbolic spaces

Let E=K(n,m,D) be a Bedford-McMullen carpet with expanding factors n,m and digit set D. We call a=(aj)j=0m−1 the distribution sequence of D whereaj=#{i;(i,j)∈D}. Under a certain vertical separation condition, Li et al. (2013) [7] showed that if two totally disconnected Bedford-McMullent carpets share the same distribution sequence, then they are Lipschitz equivalent. In this paper, we define a metric ρ on D∞ and call (D∞,ρ) a half-symbolic space. We show that if E is totally disconnected and satisfies the vertical separation condition, then E is Lipschitz equivalent to (D∞,ρ). Thanks to this result, we extend the result of Li et al. by showing that two Bedford-McMullen carpets K(n,m,D) and K(n,m,D′) are Lipschitz equivalent if a, the distribution sequence of D, is a permutation of a′, the distribution sequence of D′, and that a and a′ satisfy a certain color matchable condition.

Fourier bases of the planar self‐affine measures with three digits

AbstractFor an expansive real matrix and a noncollinear integer digit set with , let be the self‐affine measure defined by . In this paper, some necessary and sufficient conditions for contains an infinite orthogonal exponential set or to be a spectral measure are given.

Self-similar measures with product-form digit sets and their spectra

Let t≥2 be an integer and p,q be two prime numbers, D=εp⊕(pq)l−1pεq be a product-form digit set with l≥1. Let μt,D be the self-similar measure generated by the pair (t,D). It is known that Fu et al. [4,17] characterized the spectrality of μt,D. In this paper, given a spectrum Λ of μt,D, we study some integers b such that bΛ is also a spectrum of μt,D.