On inhomogeneous diophantine approximation and Hausdorff dimension

Let Γ = Z A + Z ⁿ  ⊂ R ⁿ be a dense subgroup of rank n + 1 and let [^(w)] \hat{w} (A) denote the exponent of uniform simultaneous rational approximation to the generating point A. For any real number v ≥  [^(w)] \hat{w} (A), the Hausdorff dimension of the set B _v of points in R ⁿ that are v-approximable with respect to Γ is shown to be equal to 1/v.     

On the Hausdorff dimension of fibres

It is well known that there are planar sets of Hausdorff dimension greater than 1 which are graphs of functions, i.e., all their vertical fibres consist of 1 point. We show this phenomenon does not occur for sets constructed in a certain “regular” fashion. Specifically, we consider sets obtained by partitioning a square into 4 subsquares, discarding 1 of them and repeating this on each of the 3 remaining squares, etc.; then almost all vertical fibres of a set so obtained have Hausdorff dimension at least 1/2. Sharp bounds on the dimensions of sets of exceptional fibres are presented. Partially supported by a grant from the Landau Centre for Mathematical Analysis.     

Transfinite Hausdorff dimension

Making extensive use of small transfinite topological dimension trind, we ascribe to every metric space X an ordinal number (or −1 or Ω) tHD(X), and we call it the transfinite Hausdorff dimension of X. This ordinal number shares many common features with Hausdorff dimension. It is monotone with respect to subspaces, it is invariant under bi-Lipschitz maps (but in general not under homeomorphisms), in fact like Hausdorff dimension, it does not increase under Lipschitz maps, and it also satisfies the intermediate dimension property (Theorem 2.7). The primary goal of transfinite Hausdorff dimension is to classify metric spaces with infinite Hausdorff dimension. Indeed, if tHD(X)?ω₀, then HD(X)=+∞. We prove that tHD(X)?ω₁ for every separable metric space X, and, as our main theorem, we show that for every ordinal number α<ω₁ there exists a compact metric space Xα (a subspace of the Hilbert space l₂) with tHD(Xα)=α and which is a topological Cantor set, thus of topological dimension 0. In our proof we develop a metric version of Smirnov topological spaces and we establish several properties of transfinite Hausdorff dimension, including its relations with classical Hausdorff dimension.     

On the Hausdorff dimension of the mather quotient

Under appropriate assumptions on the dimension of the ambient manifold and the regularity of the Hamiltonian, we show that the Mather quotient is small in term of the Hausdorff dimension. Then we present applications in dynamics. © 2008 Wiley Periodicals, Inc.     

On the Hausdorff dimension of rademacher Riesz products

We give a formula for the Hausdorff dimension of fractals which are the support of certain Riesz-product type measures.     

Hausdorff dimension and distance sets

According to a result of K. Falconer (1985), the setD(A)={|x−y|;x, y ∈A} of distances for a Souslin setA of ℝ ⁿ has positive 1-dimensional measure provided the Hausdorff dimension ofA is larger than (n+1)/2.* We give an improvement of this statement in dimensionsn=2,n=3. The method is based on the fine theory of Fourier restriction phenomena to spheres. Variants of it permit further improvements which we don’t plan to describe here. This research originated from some discussions with P. Mattila on the subject. dimA >n/2 would be the optimal result forn ≥ 2.     

Hausdorff dimension and asymptotic cycles

We carry out a multifractal analysis for the asymptotic cycles for compact Riemann surfaces of genus . This describes the set of unit tangent vectors for which the associated orbit has a given asymptotic cycle in homology.

     

Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems

For a sequence of integers {a(x)} _x≥1 we show that the distribution of the pair correlations of the fractional parts of {〈αa(x)〉} _x≥1 is asymptotically Poissonian for almost all α if the additive energy of truncations of the sequence has a power savings improvement over the trivial estimate. Furthermore, we give an estimate for the Hausdorff dimension of the exceptional set as a function of the density of the sequence and the power savings in the energy estimate. A consequence of these results is that the Hausdorff dimension of the set of α such that {〈αx ^d 〉} fails to have Poissonian pair correlation is at most \(\frac{{d + 2}}{{d + 3}} < 1\). This strengthens a result of Rudnick and Sarnak which states that the exceptional set has zero Lebesgue measure. On the other hand, classical examples imply that the exceptional set has Hausdorff dimension at least \(\frac{2}{{d + 1}}\).An appendix by Jean Bourgain was added after the first version of this paper was written. In this appendix two problems raised in the paper are solved.     

Univoque bases and Hausdorff dimension

Given a positive integer M and a real number \(q >1\), a q -expansion of a real number x is a sequence \((c_i)=c_1c_2\ldots \) with \((c_i) \in \{0,\ldots ,M\}^\infty \) such that

$$\begin{aligned} x=\sum _{i=1}^{\infty } c_iq^{-i}. \end{aligned}$$

It is well known that if \(q \in (1,M+1]\), then each \(x \in I_q:=\left[ 0,M/(q-1)\right] \) has a q-expansion. Let \(\mathcal {U}=\mathcal {U}(M)\) be the set of univoque bases \(q>1\) for which 1 has a unique q-expansion. The main object of this paper is to provide new characterizations of \(\mathcal {U}\) and to show that the Hausdorff dimension of the set of numbers \(x \in I_q\) with a unique q-expansion changes the most if q “crosses” a univoque base. Denote by \(\mathcal {B}_2=\mathcal {B}_2(M)\) the set of \(q \in (1,M+1]\) such that there exist numbers having precisely two distinct q-expansions. As a by-product of our results, we obtain an answer to a question of Sidorov (J Number Theory 129:741–754, 2009) and prove that

$$\begin{aligned} \dim _H(\mathcal {B}_2\cap (q',q'+\delta ))>0\quad \text {for any}\quad \delta >0, \end{aligned}$$

where \(q'=q'(M)\) is the Komornik–Loreti constant.

     

Prime-representing functions and Hausdorff dimension

Acta Mathematica Hungarica - Let $$c \geq 2$$ be any fixed real number. Matomäki [4] inverstigated the set of $$A &gt; 1$$ such that the integer part of $$ A^{c^k}$$ is a prime number for...     

Local self-similarity and the Hausdorff dimension

Let X be a locally self-similar stochastic process of index 0<H<1 whose sample paths are a.s. C^H?ε for all ε>0. Then the Hausdorff dimension of the graph of X is a.s. 2?H. To cite this article: A. Benassi et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Hausdorff dimension and dendritic limit sets

Let be a singly degenerate closed surface group acting properly discontinuously on hyperbolic 3-space, H³, such that H³/ has positive injectivity radius. It is known that the limit set is a dendrite of Hausdorff dimension 2. We show that the cut-point set of the limit set has Hausdorff dimension strictly less than 2.     

Hausdorff dimension and hierarchical system dynamics

We show that the Hausdorff dimension may be used to distinguish different relaxation dynamics in hierarchical systems. We examine the hierarchical systems following the temperature-dependent power-law decay and the Kohlrausch law. For our purposes, we consider random walks on p-adic integer numbers.     

On Hausdorff dimension of certain Riesz product in local fields

In this paper, we consider the Riesz product dμ =^∞∏j=1（1＋ajRexbjλj（x））dx in local fields, and we obtain the upper and lower bound of its Hausdorff dimension.