The Spectral Properties of the Strongly Coupled Sturm Hamiltonian of Eventually Constant Type

We study the spectral properties of the Sturm Hamiltolian of eventually constant type, which includes the Fibonacci Hamiltonian. Let s be the Hausdorff dimension of the spectrum. For V > 20, we show that the restriction of the s-dimensional Hausdorff measure to the spectrum is a Gibbs type measure; the density of states measure is a Markov measure. Based on the fine structures of these measures, we show that both measures are exact dimensional; we obtain exact asymptotic behaviors for the optimal Hölder exponent and the Hausdorff dimension of the density of states measure and for the Hausdorff dimension of the spectrum. As a consequence, if the frequency is not silver number type, then for V big enough, we establish strict inequalities between these three spectral characteristics. We achieve them by introducing an auxiliary symbolic dynamical system and applying the thermodynamical and multifractal formalisms of almost additive potentials.     

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.     

A sharp bound on the Hausdorff dimension of the singular set of a uniform measure

The study of the geometry of n-uniform measures in \(\mathbb {R}^{d}\) has been an important question in many fields of analysis since Preiss’ seminal proof of the rectifiability of measures with positive and finite density. The classification of uniform measures remains an open question to this day. In fact there is only one known example of a non-trivial uniform measure, namely 3-Hausdorff measure restricted to the Kowalski–Preiss cone. Using this cone one can construct an n-uniform measure whose singular set has Hausdorff dimension \(n-3\). In this paper, we prove that this is the largest the singular set can be. Namely, the Hausdorff dimension of the singular set of any n-uniform measure is at most \(n-3\).     

Random closed sets viewed as random recursions

It is known that the box dimension of any Martin-Löf random closed set of ${\{0,1\}^\mathbb{N}}$ is ${\log_2(\frac{4}{3})}$ . Barmpalias et al. [J Logic Comput 17(6):1041–1062, 2007] gave one method of producing such random closed sets and then computed the box dimension, and posed several questions regarding other methods of construction. We outline a method using random recursive constructions for computing the Hausdorff dimension of almost every random closed set of ${\{0,1\}^\mathbb{N}}$ , and propose a general method for random closed sets in other spaces. We further find both the appropriate dimensional Hausdorff measure and the exact Hausdorff dimension for such random closed sets.     

The dimensions of the divergence points of self-similar measures with weak separation condition

Let \(\mu \) be the self-similar measure supported on the self-similar set K with the weak separation condition, which is weaker than the open set condition. This article uses Hausdorff dimension and packing dimension to investigate the multifractal structure of several sets of divergence points of \(\mu \) in the iterated function system.     

Statistically self-affine sets: Hausdorff and box dimensions

We introduce a class of random sets in ₂ that include McMullen's generalized Sierpinski carpets. We give exact expressions for the Hausdorff and Bouligand-Minkowski (box) dimensions of these sets, and find in particular that typically they are not equal. Our expression for the Hausdorff dimension isnot what one would expect by analogy with McMullen's formula for the Hausdorff dimension of a generalized Sierpinski carpet.Supported by Davis Ross Grant.Supported by NSF Grant DMS-9146252. Department of Statistics, Purdue University.     

Random cutout sets with spatially inhomogeneous intensities

We study the Hausdorff dimension of Poissonian cutout sets defined via inhomogeneous intensity measures on Q-regular metric spaces. Our main results explain the dependence of the dimension of the cutout sets on the multifractal structure of the average densities of the Q-regular measure. As a corollary, we obtain formulas for the Hausdorff dimension of such cutout sets in self-similar and self-conformal spaces using the multifractal decomposition of the average densities for the natural measures.     

Squares and their centers

We study the relationship between the size of two sets B, S ? R², when B contains either the whole boundary or the four vertices of a square with axes-parallel sides and center in every point of S. Size refers to cardinality, Hausdorff dimension, packing dimension, or upper or lower box dimension. Perhaps surprisingly, the results vary depending on the notion of size under consideration. For example, we construct a compact set B of Hausdorff dimension 1 which contains the boundary of an axes-parallel square with center in every point in [0, 1]², prove that such a B must have packing and lower box dimension at least 7/4, and show by example that this is sharp. For more general sets of centers, the answers for packing and box counting dimensions also differ. These problems are inspired by the analogous problems for circles that were investigated by Bourgain, Marstrand and Wolff, among others.     

The Integral Formula for Calculating the Hausdorff Measure of Some Fractal Sets

It is important to calculate the Hausdorff dimension and the Hausdorff mesure respect to this dimension for some fractal sets. By using the usual method of Mass Distribution, we can only calculate the Hausdorff dimension. In this paper, we will construct an integral formula by using lower inverse s-density and then use it to calculate the Hausdorff measures for some fractional dimensional sets.     

Some dimensional results for a class of special homogeneous Moran sets

We construct a class of special homogeneous Moran sets, called {m_k}-quasi homogeneous Cantor sets, and discuss their Hausdorff dimensions. By adjusting the value of {m_k}_k?1, we constructively prove the intermediate value theorem for the homogeneous Moran set. Moreover, we obtain a sufficient condition for the Hausdorff dimension of ho- mogeneous Moran sets to assume the minimum value, which expands earlier works.     

Inductive topological Hausdorff dimensions and fibers of generic continuous functions

In an earlier paper Buczolich, Elekes and the author introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. They proved that it is precisely the right notion to describe the Hausdorff dimension of the level sets of the generic real-valued continuous function (in the sense of Baire category) defined on a compact metric space $K$ . The goal of this paper is to determine the Hausdorff dimension of the fibers of the generic continuous function from $K$ to $\mathbb {R}^n$ . In order to do so, we define the $n$ th inductive topological Hausdorff dimension, $\dim _{t^nH} K$ . Let $\dim _H K,\,\dim _t K$ and $C_n(K)$ denote the Hausdorff and topological dimension of $K$ and the Banach space of the continuous functions from $K$ to $\mathbb {R}^n$ . We show that $\sup _{y\in \mathbb {R}^n} \dim _{H}f^{-1}(y) = \dim _{t^nH} K -n$ for the generic $f \in C_n(K)$ , provided that $\dim _t K\ge n$ , otherwise every fiber is finite. In order to prove the above theorem we give some equivalent definitions for the inductive topological Hausdorff dimensions, which can be interesting in their own right. Here we use techniques coming from the theory of topological dimension. We show that the supremum is actually attained on the left hand side of the above equation. We characterize those compact metric spaces $K$ for which $\dim _{H} f^{-1}(y)=\dim _{t^nH}K-n$ for the generic $f\in C_n(K)$ and the generic $y\in f(K)$ . We also generalize a result of Kirchheim by showing that if $K$ is self-similar and $\dim _t K\ge n$ then $\dim _{H} f^{-1}(y)=\dim _{t^nH}K-n$ for the generic $f\in C_n(K)$ for every $y\in {{\mathrm{int}}}f(K)$ .     

Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise

We consider a system of d non-linear stochastic heat equations in spatial dimension 1 driven by d-dimensional space-time white noise. The non-linearities appear both as additive drift terms and as multipliers of the noise. Using techniques of Malliavin calculus, we establish upper and lower bounds on the one-point density of the solution u(t, x), and upper bounds of Gaussian-type on the two-point density of (u(s, y),u(t, x)). In particular, this estimate quantifies how this density degenerates as (s, y) → (t, x). From these results, we deduce upper and lower bounds on hitting probabilities of the process ${\{u(t,x)\}_{t \in \mathbb{R}_+, x\in [0,1]}}$ , in terms of respectively Hausdorff measure and Newtonian capacity. These estimates make it possible to show that points are polar when d ≥ 7 and are not polar when d ≤ 5. We also show that the Hausdorff dimension of the range of the process is 6 when d > 6, and give analogous results for the processes ${t \mapsto u(t,x)}$ and ${x \mapsto u(t,x)}$ . Finally, we obtain the values of the Hausdorff dimensions of the level sets of these processes.     

On the Hausdorff volume in sub-Riemannian geometry

For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, while starting from dimension 5, in corank 1 case, it is ${\mathcal {C}^{3}}$ (and ${\mathcal {C}^{4}}$ on every smooth curve) but in general not ${\mathcal {C}^{5}}$ . These results answer to a question addressed by Montgomery about the relation between two intrinsic volumes that can be defined in a sub-Riemannian manifold, namely the Popp and the Hausdorff volume. If the nilpotent approximation depends on the point (that may happen starting from dimension 5), then they are not proportional, in general.     

On the Distribution Modulo 1 of Exponential Sequences

New quantitative results on the intersection of winning sets and the Hausdorff dimension of this intersection are obtained. An application to the problem on fractional parts of the sequence $\{ 2^n 3^m \alpha \}$ is given.     

On the Hausdorff dimension of the inverse image of a compact set under a levy process

In this paper, we obtain some formulas for determining the Hausdorff dimension of the inverse image of compact sets under general Levy processes in ⁿ .Supported by the British Council.     

Dimensions of fibers of generic continuous maps

In an earlier paper Buczolich, Elekes, and the author described the Hausdorff dimension of the level sets of a generic real-valued continuous function (in the sense of Baire category) defined on a compact metric space K by introducing the notion of topological Hausdorff dimension. Later on, the author extended the theory for maps from K to \({\mathbb {R}}^n\). The main goal of this paper is to generalize the relevant results for topological and packing dimensions and to obtain new results for sufficiently homogeneous spaces K even in the case case of Hausdorff dimension. Let K be a compact metric space and let us denote by \(C(K,{\mathbb {R}}^n)\) the set of continuous maps from K to \({\mathbb {R}}^n\) endowed with the maximum norm. Let \(\dim _{*}\) be one of the topological dimension \(\dim _T\), the Hausdorff dimension \(\dim _H\), or the packing dimension \(\dim _P\). Define

$$\begin{aligned} d_{*}^n(K)=\inf \left\{ \dim _{*}(K{\setminus } F): F\subset K \text { is } \sigma \text {-compact with } \dim _T F<n\right\} . \end{aligned}$$

We prove that \(d^n_{*}(K)\) is the right notion to describe the dimensions of the fibers of a generic continuous map \(f\in C(K,{\mathbb {R}}^n)\). In particular, we show that \(\sup \{\dim _{*}f^{-1}(y): y\in {\mathbb {R}}^n\} =d^n_{*}(K)\) provided that \(\dim _T K\ge n\), otherwise every fiber is finite. Proving the above theorem for packing dimension requires entirely new ideas. Moreover, we show that the supremum is attained on the left hand side of the above equation. Assume \(\dim _T K\ge n\). If K is sufficiently homogeneous, then we can say much more. For example, we prove that \(\dim _{*}f^{-1}(y)=d^n_{*}(K)\) for a generic \(f\in C(K,{\mathbb {R}}^n)\) for all \(y\in {{\mathrm{int}}}f(K)\) if and only if \(d^n_{*}(U)=d^n_{*}(K)\) or \(\dim _T U<n\) for all open sets \(U\subset K\). This is new even if \(n=1\) and \(\dim _{*}=\dim _H\). It is known that for a generic \(f\in C(K,{\mathbb {R}}^n)\) the interior of f(K) is not empty. We augment the above characterization by showing that \(\dim _T \partial f(K)=\dim _H \partial f(K)=n-1\) for a generic \(f\in C(K,{\mathbb {R}}^n)\). In particular, almost every point of f(K) is an interior point. In order to obtain more precise results, we use the concept of generalized Hausdorff and packing measures, too.

     

Removable sets for homogeneous linear partial differential equations in Carnot groups

Let L be a homogeneous left-invariant differential operator on a Carnot group. Assume that both L and L^t are hypoelliptic. We study the removable sets for L-solutions. We give precise conditions in terms of the Carnot- Caratheodory Hausdorff dimension for the removability for L-solutions under several auxiliary integrability or regularity hypotheses. In some cases, our criteria are sharp on the level of the relevant Hausdorff measure. One of the main ingredients in our proof is the use of novel local self-similar tilings in Carnot groups.     

Hausdorff Dimension of Harmonic Measure for Self-Conformal Sets

Under some technical assumptions it is shown that the Hausdorff dimension of the harmonic measure on the limit set of a conformal infinite iterated function system is strictly less than the Hausdorff dimension of the limit set itself if the limit set is contained in a real-analytic curve, if the iterated function system consists of similarities only, or if this system is irregular. As a consequence of this general result the same statement is proven for hyperbolic and parabolic Julia sets, finite parabolic iterated function systems and generalized polynomial-like mappings. Also sufficient conditions are provided for a limit set to be uniformly perfect and for the harmonic measure to have the Hausdorff dimension less than 1. Some results in the spirit of Przytycki et al. (Ann. of Math.130 (1989), 1-40; Stud. Math.97 (1991), 189-225) are obtained.     

A note on the quantization for probability measures with respect to the geometric mean error

We study the quantization with respect to the geometric mean error for probability measures μ on \({\mathbb{R}^d}\) for which there exist some constants C, η > 0 such that \({\mu(B(x,\varepsilon))\leq C\varepsilon^\eta}\) for all ε > 0 and all \({x\in\mathbb{R}^d}\) . For such measures μ, we prove that the upper quantization dimension \({\overline{D}(\mu)}\) of μ is bounded from above by its upper packing dimension and the lower one \({\underline{D}(\mu)}\) is bounded from below by its lower Hausdorff dimension. This enables us to calculate the quantization dimension for a large class of probability measures which have nice local behavior, including the self-affine measures on general Sierpiński carpets and self-conformal measures. Moreover, based on our previous work, we prove that the upper and lower quantization coefficient for a self-conformal measure are both positive and finite.     

Regularity of generalized solutions to the neumann problem for higher-order equations

It is proved that a generalized solution to the Neumann problem for a nonlinear higher-order equation belongs to the Hölder space C^m?1,α in the set $\bar \Omega \backslash \Sigma $ for a dimension n≤4. It is shown that the set Σ is empty for n=2. In the case n=3.4 the estimate of the Hausdorff dimension of the set E is obtained. Bibliography: 19 titles.