Hausdorff Dimension Of Certain Sets Research Articles

Hausdorff dimension of certain sets related to random αβ-orbits which are not dense

Let α,β∈[0,1)=R/Z such that at least one of them is irrational. It is known that the random αβ-orbits of the Bernoulli mixture of rotations of α and β by choosing them with equal probability 12 are uniformly distributed modulo 1 with probability one. Chen et al. (2021) [2] showed that the exceptional set in the probability space has full Hausdorff dimension. In this note, we prove that certain sets related to random αβ-orbits which are not dense in R/Z have Hausdorff dimension 1. Our result can be viewed as an improvement of the dimension result obtained by Chen, Wang and Wen.

Hausdorff dimension of certain sets arising by the maximal run-length function over factorial language

Let A be a language with infinite number of words over the alphabet Σ={0,1} and let (xi)i=1+∞ be the digits sequence in the unique terminating dyadic expansion of x∈[0,1). The maximal run-length function ln(x,A) of x over A is defined byln(x,A)=max⁡{k:xi+1xi+2⋯xi+k∈Afor some0≤i≤n−k}. In a general setting, Wu (2024) [18] proved a metrical property on the asymptotic behavior of ln(x,A), which extends the famous Erdős-Rényi limit theorem that represents for the special case A={1k:k≥1}. Under the assumption that A is factorial (i.e. the language A is closed under subword), Wu studied the exceptional setE={x∈[0,1):limn→∞⁡ln(x,A)mn=0andlimn→∞⁡ln(x,A)=∞}, and he obtained a lower bound of the Hausdorff dimension of E with some conditions on mn. In this paper, we determine the Hausdorff dimension of E without any additional assumption on mn and we obtain the Hausdorff dimension of two other kinds of exceptional sets of elements whose maximal run-length functions have different growth rates. Our results significantly extend the existing dimension results in this topic.

Topological Pressure of Free Semigroup Actions For Non-Compact Sets and Bowen’s Equation, II

Inspired to the work of Ma and Wu (Discrete Contin Dyn Syst 31:545–557, 2011) and Climenhaga (Ergodic Theory Dyn Syst 31(4):1163–1182, 2011), we introduce the new notions of topological pressure and upper (lower) capacity topological pressure of a free semigroup action of maps for finite potential functions by using the Carathéodory–Pesin structure (C-P structure) with respect to non-compact sets in this paper. Meanwhile, we also give some properties of these notions. Moreover, by Bowen’s equation, we characterize the Hausdorff dimension of certain sets, whose points have the positive lower Lyapunov exponents and satisfy a so-called tempered contraction condition.

Hausdorff dimension of certain sets arising in Engel expansions

The present paper is concerned with the Hausdorff dimension of certain sets arising in Engel expansions. In particular, the Hausdorff dimension of the set is completely determined, where An(x) can stand for the digit, gap and ratio between two consecutive digits in the Engel expansion of x and ϕ is a positive function defined on natural numbers. These results significantly extend the existing results of Galambos’ open problems on the Hausdorff dimension of sets related to the growth rate of digits.

Hausdorff dimension of certain sets arising in Engel continued fractions

In the present paper, we are concerned with the Hausdorff dimension of certain sets arising in Engel continued fractions. In particular, the Hausdorff dimension of sets \begin{document}$\big\{x ∈ [0,1): b_n(x) ≥ \phi (n)~i.m.~n ∈ \mathbb{N}\big\}\ \ \text{and}\ \ \big\{x ∈ [0,1): b_n(x) ≥ \phi(n),\ \forall n ≥ 1\big\}$ \end{document} are completely determined, where $i.m.$ means infinitely many, $\{b_n(x)\}_{n ≥ 1}$ is the sequence of partial quotients of the Engel continued fraction expansion of $x$ and $\phi$ is a positive function defined on natural numbers.

On Hausdorff Dimension of Certain Sets Arising from Diophantine Approximations for Complex Numbers

We discuss the Hausdorff dimension of certain sets related to Diophantine approximations over an imaginary quadratic field $\mathbb{Q}(\sqrt{d})$. We show that, for an infinite subset $\mathcal{A}$ of $\mathbb{Z}[\omega] \backslash \{0\}$, the set of $z \in \mathbb{C}$ with $|z-a/r| 0$ has Hausdorff dimension $2(1+\gamma) / (1+\rho)$, where $\gamma$ is the sup of $h$ such that $\sum_{r \in \mathcal{A}} 1/(|r|^{2})^{h}$ diverges. This result is a version of a result by G. Harman for complex numbers without the coprime condition. In particular, this result implies a version of the classical Jarnik-Besicovitch result when we take $\mathcal{A} = \mathbb{Z}[\omega]\backslash\{0\}$. We also discuss the Hausdorff dimension of the set of complex numbers which have infinitely many solutions to the Diophantine inequality concerning the Duffin-Schaeffer conjecture over $\mathbb{Q}(\sqrt{d})$.

The Hausdorff dimension of certain sets in a class of α-Lüroth expansions

In this paper, we consider sets of points with some restricts on the digits of their α-Luroth expansions. More precisely, for any countable partition α = {A n , n ∈ ℕ} of the unit interval I, we completely determine the Hausdorff dimensions of the sets $$F\left( {\alpha ,\varphi } \right) = \left\{ {x = \left[ {l_1 \left( x \right),l_2 \left( x \right), \ldots } \right]_\alpha \in I:l_n \left( x \right) \geqslant \varphi \left( n \right),\forall _n \geqslant 1} \right\},$$ where φ is an arbitrary positive function defined on ℕ satisfying φ(n) → ∞ as n → ∞.

Bowen’s equation in the non-uniform setting

AbstractWe show that Bowen’s equation, which characterizes the Hausdorff dimension of certain sets in terms of the topological pressure of an expanding conformal map, applies in greater generality than has been heretofore established. In particular, we consider an arbitrary subset Z of a compact metric space and require only that the lower Lyapunov exponents be positive on Z, together with a tempered contraction condition. Among other things, this allows us to compute the dimension spectrum for Lyapunov exponents for maps with parabolic periodic points, and to relate the Hausdorff dimension to the topological entropy for arbitrary subsets of symbolic space with the appropriate metric.

Hausdorff dimension of certain sets arising in continued fraction expansions

This paper is concerned with the fractional dimensions of some sets of points with their partial quotients obeying some restrictions in their continued fraction expansions. The Hausdorff dimension of the following set, which shares a dichotomy law according to Borel–Bernstein's theorem, is completely determined E ( ϕ ) = { x ∈ [ 0 , 1 ) : a n ( x ) ⩾ ϕ ( n ) , i.o. n } . Our result significantly strengthens the results of Good and Lúczak.

DISTRIBUTION OF DIGITS IN INTEGERS: BESICOVITCH–EGGLESTON SUBSETS OF ${\mathbb N}$

Fix an integer N ⩾ 2. For a positive integer n ∈ N, let n = d0(n) + d1(n)N + d2(n)N2+…+dγ(n)(n)Nγ(n) where di(n) ∈ {0,1,2,…,N − 1} and dγ(n)(n) ≠ 0 denote the N-ary expansion of n. For a probability vector p = (p0…,pN−1) and r > 0, the r approximative discrete Besicovitch–Eggleston set Br(p) is defined by B r ( p ) = { ∈ N | | | { 0 ⩽ k ⩽ γ ( n ) | d k ( n ) = i } | γ ( n ) + 1 − p i | ⩽ r forall i } , that is, Br(p) is the set of positive integers n such that the frequency of the digit i in the N-ary expansion of n differs from pi by less than r for all i∈ {0,1,2,…,N − 1}. Three natural fractional dimensions of subsets E of N are defined, namely, the lower fractional dimension dim ¯ ( E ) , the upper fractional dimension dim ¯ ( E ) and the exponent of convergence δ(E), and the dimensions of various subsets of N defined in terms of the frequencies of the digits in the N-ary expansion of the positive integers are studied. In particular, the dimensions of Br(p) are computed (in the limit as rs 0). Let p=(p0,…,pN−1) be a probability vector. Then lim r → 0 dim ⇀ ( B r ( p ) ) = lim r → 0 dim ¯ ( B r ( p ) ) = lim r → 0 δ ( B r ( p ) ) = − ∑ i p i log p i log N This result provides a natural discrete analogue of a classical result due to Besicovitch and Eggleston on the Hausdorff dimension of certain sets of non-normal numbers. Several applications to the theory of normal numbers are given.

Uniform Distribution and Fractal Dimensions for Generalized Cantor Products By

For any \(\) and \(\), let \(\) be a generalized Cantor product. The Hausdorff dimension of certain sets concerning \(\) are considered. Let \(\) be defined as in [11], the exceptional set of values of x for which \(\) is not uniformly distributed modulo 1 is also determined.

The Hausdorff dimension of certain sets arising from Diophantine approximation by restricted sequences of integer vectors

The Hausdorff dimension of certain sets arising from Diophantine approximation by restricted sequences of integer vectors

The Hausdorff dimension of certain sets of singular n-tuples

In the paper [3] we obtained a lower bound for the Hausdorff dimension of certain sets of singular n-tuples in ℝn. In this paper we will obtain an upper bound for the dimension of these sets. These two bounds are not sharp enough to yield the exact dimension of all the sets considered but they do yield the dimension of the set of ‘highly singular’ (in a sense to be made precise below) n-tuples.

A lower bound for the Hausdorff dimension of sets of singular n-tuples

In this paper we improve the estimate for the lower bound of the Hausdorff dimension of certain sets of singular n-tuples in ℝn obtained by Baker in [1]. We begin with a brief discussion of the problem. More details can be found in [1]. For ease of comparison with Baker's results we will adopt a similar notation to that of [1].

The Hausdorff dimension of certain sets of nonnormal numbers.

The Hausdorff dimension of certain sets of nonnormal numbers.