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The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number $\alpha$ and a convex body $B$, $g_{\alpha}(B)$ is the infimum of $\alpha$-powers of finitely many homothety coefficients less than 1 such that there is a covering of $B$ by translative homothets with these coefficients. $h_{\alpha}(B)$ is the minimal number of directions such that the boundary of $B$ can be illuminated by this number of directions except for a subset whose Hausdorff dimension is less than $\alpha$. In this paper, we prove that $g_{\alpha}(B)\leq h_{\alpha}(B)$, find upper and lower bounds for both numbers, and discuss several general conjectures. In particular, we show that $h_{\alpha} (B) > 2^{d-\alpha}$ for almost all $\alpha$ and $d$ when $B$ is the $d$-dimensional cube, thus disproving the conjecture from Research Problems in Discrete Geometry by Brass, Moser, and Pach.

In this paper we show that given any compact set $E \subset \hat{\mathbb{C}}$, we can always find a conformally removable subset with the same Hausdorff dimension as $E$.

We prove that for any compact set E ⊂ R 2 E\subset \mathbb {R}^2 , dim H ⁡ ( E ) > 1 \dim _{\mathcal {H}}(E)>1 , there exists x ∈ E x\in E such that the Hausdorff dimension of the pinned distance set Δ x ( E ) = { | x − y | : y ∈ E } \begin{equation*} \Delta _x(E)=\{|x-y|: y \in E\} \end{equation*} is no less than min { 4 3 dim H ⁡ ( E ) − 2 3 , 1 } \min \left \{\frac {4}{3}\dim _{\mathcal {H}}(E)-\frac {2}{3}, 1\right \} . This answers a question recently raised by Guth, Iosevich, Ou, and Wang, as well as improves results of Keleti and Shmerkin.

We study subsets of $\R^{d}$ which are thin for doubling measures or isotropic doubling measures. We show that any subset of $\R^{d}$ with Hausdorff dimension less than or equal to $d-1$ is thin for isotropic doubling measures. We also prove that a self-affine set that satisfies $OSCH$ (open set condition with holes) is thin for isotropic doubling measures. For doubling measures, we prove that Bara\'nski carpets are thin for doubling measures., Comment: 16pages, 4 figures

In this paper, we consider the system $-\Delta u =\lambda (v+1)^p,\;\;-\Delta v = \gamma (u+1)^\theta$ on a smooth bounded domain $\Omega$ in $\mathbb{R}^N$ with the Dirichlet boundary condition $u=v=0$ on $\partial \Omega.$ Here $ \lambda,\gamma$ are positive parameters. Let $x_0$ be the largest root of the polynomial \begin{equation*} H(x) = x^4 - \frac{16p\theta(p+1)(\theta+1)}{(p\theta-1)^2}x^2 + \frac{16p\theta(p+1)(\theta+1)(p+\theta+2)}{(p\theta-1)^3}x -\frac{16p\theta(p+1)^2(\theta+1)^2}{(p\theta-1)^4}. \end{equation*} We show that the extremal solutions associated to the above system are bounded provided $N

We define a Bishop-type inequality on metric measure spaces with Riemannian curvature-dimension condition. The main result in this short article is that any R C D RCD spaces with the Bishop-type inequalities possess only one regular set in not only the measure theoretical sense but also the set theoretical one. As a corollary, the Hausdorff dimension of such R C D ∗ ( K , N ) RCD^*(K,N) spaces is exactly N N . We also prove that every tangent cone at any point on such R C D RCD spaces is a metric cone.

I. J. Good (1941) showed that the set of irrational numbers in $(0,1)$ whose partial quotients $a_n$ tend to infinity is of Hausdorff dimension $1/2$. A number of related results impose restrictions of the type $a_n\in B$ or $a_n\geq f(n)$, where $B$ is an infinite subset of $\mathbb N$ and $f$ is a rapidly growing function with $n$. We show that, for an arbitrary $B$ and an arbitrary $f$ with values in $[\min B,\infty)$ and tending to infinity, the set of irrational numbers in $(0,1)$ such that \[ a_n\in B,\ a_n\leq f(n)\text{ for all $n\in\mathbb N$, and }a_n\to\infty\text{ as }n\to\infty\] is of Hausdorff dimension $\tau(B)/2,$ where $\tau(B)$ is the exponent of convergence of $B$., Comment: to appear in Proc. Amer. Math. Soc

We apply the theory of Peres and Schlag to obtain estimates for generic Hausdorff dimension distortion under orthogonal projections on simply connected two dimensional Riemannian manifolds of constant curvature. As a conclusion we obtain appropriate versions of Marstrand theorem, Kaufman's theorem and Falconer's theorem in the above geometrical settings., 11 pages. Version 2: corrected typos, added references, changed title, improved structure in Section 3. This corresponds to the published version

We extend a Poincare-type inequality for functions with large zero-sets by Jiang and Lin to fractional Sobolev spaces. As a consequence, we obtain a Hausdorff dimension estimate on the size of zero sets for fractional Sobolev functions whose inverse is integrable. Also, for a suboptimal Hausdorff dimension estimate, we give a completely elementary proof based on a pointwise Poincare-style inequality.

Let H h \mathcal {H}^h be the h h -dimensional Hausdorff measure on R d \mathbb {R}^d . Besicovitch showed that if a set E E is null for H h \mathcal {H}^h , then it is null for H g \mathcal {H}^g , for some dimension g g smaller than h h . We prove that this is not true for packing measures. Moreover, we consider the corresponding questions for sets of non- σ \sigma -finite packing measure and for pre-packing measure instead of packing measure.

A power series f(z) which converges in D = {|z| < 1} maps the radii [0, ζ) onto paths Γ(ζ), ζ ∈ T = ∂D. These are studied under several aspects in the case of the special lacunary series f(z) = z+ z2 + z4 + z8 + . . .. First the Γ(ζ) are considered as random functions on the probability space (T,B,mes /2π), where B is the σ-algebra of Borel sets and mes the Lebesgue measure. Then analytical properties of the Γ(ζ) are discussed which hold on subsets A of T with Hausdorff dimension 1 in spite of mesA = 0. Furthermore, estimates of the derivative of f and of the arc length of sections of the Γ(ζ) are given. Finally, these results are used to derive connections between the distribution of critical points of f and the overall behaviour of the paths.

We investigate the set of biaccessible points for connected polynomial Julia sets of arbitrary degrees d ≥ 2 d\geq 2 . We prove that the Hausdorff dimension of the set of external angles corresponding to biaccessible points is less than 1 1 , unless the Julia set is an interval. This strengthens theorems of Stanislav Smirnov and Anna Zdunik: they proved that the same set of external angles has zero 1 1 -dimensional measure.

JMF was supported by an EPSRC grant whilst undertaking this work. Given a compact subset F of R2, the visible part VθF of F from direction θ is the set of x in F such that the half-line from x in direction θ intersects F only at x. It is suggested that if dimH F ≥ 1 then dimH VθF = 1 for almost all θ , where dimH denotes Hausdorff dimension. We conrm this when F is a self-similar set satisfying the convex open set condition and such that the orthogonal projection of F onto every line is an interval. In particular the underlying similarities may involve arbitrary rotations and F need not be connected. Publisher PDF

We show that for a transcendental entire function the set of points whose orbit under iteration is bounded can have arbitrarily small positive Hausdorff dimension., 7 pages

Let $f$ and $g$ be transcendental entire functions, each with a bounded set of singular values, and suppose that $f$ and $g$ are affinely equivalent (that is, $g \circ \phi= \psi\circ f$, where $\phi,\psi:\C\to\C$ are affine). We show that the escaping sets of $f$ and $g$ have the same Hausdorff dimension. Using a result of the second author, we deduce that there exists a family of transcendental entire functions for which the escaping set has Hausdorff dimension equal to one., Comment: 9 pages

We survey the definition of the radial Julia set of a meromorphic function (in fact, more generally, any "Ahlfors islands map"), and give a simple proof that the Hausdorff dimension of the reduced Julia set always coincides with the hyperbolic dimension., Comment: 11 pages

Our main result shows that a continuous map f f acting on a compact metric space ( X , ρ ) (X,\rho ) with a weaker form of specification property and with a pair of distal points is distributionally chaotic in a very strong sense. Strictly speaking, there is a distributionally scrambled set S S dense in X X which is the union of disjoint sets homeomorphic to Cantor sets so that, for any two distinct points u , v ∈ S u,v\in S , the upper distribution function is identically 1 and the lower distribution function is zero at some ε > 0 \varepsilon >0 . As a consequence, we describe a class of maps with a scrambled set of full Lebesgue measure in the case when X X is the k k -dimensional cube I k I^{k} . If X = I X=I , then we can even construct scrambled sets whose complements have zero Hausdorff dimension.

We construct a II 0 1 -class X that has classical packing dimension 0 and effective packing dimension 1. This implies that, unlike in the case of effective Hausdorff dimension, there is no natural correspondence principle (as defined by Lutz) for effective packing dimension. We also examine the relationship between upper box dimension and packing dimension for II 0 1 -classes.

The central set of a domain D is the set of centers of maximal discs in D. Kremlin proved that the central set of a planar domain has zero area and asked whether it can have Hausdorff dimension strictly larger than 1. We construct a planar domain with central set of Hausdorff dimension 2.

Let {X(t)} t≥0 denote a Levy process in R d with exponent Ψ. Taylor (1986) proved that the packing dimension of the range X([0,1]) is given by the index (0.1) γ' = sup {α ≥ 0: lim inf r→0 + ∫ 1 0 P|X(t)|≤r r α dt=0}. We provide an alternative formulation of γ' in terms of the Levy exponent Ψ. Our formulation, as well as methods, are Fourier-analytic, and rely on the properties of the Cauchy transform. We show, through examples, some applications of our formula.

We investigate the asymptotic behavior, as N grows, of the largest minimal pairwise distance of N points restricted to an arbitrary compact rec- tifiable set embedded in Euclidean space, and we find the limit distribution of such optimal configurations. For this purpose, we compare best-packing configurations with minimal Riesz s-energy configurations and determine the s-th root asymptotic behavior (as s ! 1) of the minimal energy constants. We show that the upper and the lower dimension of a set defined through the Riesz energy or best-packing coincides with the upper and lower Minkowski dimension, respectively. For certain sets in R d of integer Hausdor dimension, we show that the limiting behavior of the best-packing distance as well as the minimal s-energy for large s is dierent for dierent subsequences of the cardinalities of the configurations.

The aim of this note is to study the construction of the boundary of a self-similar tile, which is generated by an iterated function system {Φ i (x) = 1/N(x + d i )} N i=1 . We will show that the boundary has complicated structure (no simple points) in general; however, it is a regular fractal set.

We prove an estimate for the spherical average operator in R d if d > 3. This leads to a lower bound for the Hausdorff dimension of unions of certain collections of spheres and to a Strichartz-type estimate for solutions of the wave equation.

We give examples of Cantor sets in C n \mathbb {C}^n of Hausdorff dimension 1 whose polynomial hulls have non-empty interior.

In this paper we investigate the regularity of stable-stationary harmonic maps. By assuming that the target manifolds do not carry any stable harmonic S 2 , we obtain some compactness results and regularity theorems. In particular, we prove that the Hausdorff dimension of the singular set of these maps cannot exceed n - 3, and the dimension estimate is optimal.

Let Y Y be a proximinal subspace of finite codimension of c 0 c_0 . We show that Y Y is proximinal in ℓ ∞ \ell _\infty and the metric projection from ℓ ∞ \ell _\infty onto Y Y is Hausdorff metric continuous. In particular, this implies that the metric projection from ℓ ∞ \ell _\infty onto Y Y is both lower Hausdorff semi-continuous and upper Hausdorff semi-continuous.

If G is any Kleinian group, we show that the dimension of the limit set A is always equal to either the dimension of the bounded geodesics or the dimension of the geodesics that escape to infinity at linear speed.

We show that there exists a locally compact Cantor minimal system whose topological spectrum has a given Hausdor dimension.