Fourier integral operators, fractal sets, and the regular value theorem

We prove that if E⊂R2d, for d?2, is an Ahlfors–David regular product set of sufficiently large Hausdorff dimension, denoted by dimH(E), and ? is a sufficiently regular function, then the upper Minkowski dimension of the set does not exceed dimH(E)−m, in line with the regular value theorem from the elementary differential geometry. Our arguments are based on the mapping properties of the underlying Fourier integral operators and are intimately connected with the Falconer distance conjecture in geometric measure theory. We shall see that our results are, in general, sharp in the sense that if the Hausdorff dimension is smaller than a certain threshold, then the dimensional inequality fails in a quantifiable way. The constructions used to demonstrate this are based on the distribution of lattice points on convex surfaces and have connections with combinatorial geometry.     

Hausdorff measure of uniform self-similar fractals

Let d ≥ 1 be an integer and E a self-similar fractal set, which is the attractor of a uniform contracting iterated function system (UIFS) on R ^d . Denote by D the Hausdorff dimension, by H ^D (E) the Hausdorff measure and by diam(E) the diameter of E. If the UIFS is parametrised by its contracting factor c, while the set ω of fixed points of the UIFS does not depend on c, we will show the existence of a positive constant depending only on ω, such that the Hausdorff dimension is smaller than one and H ^D (E) = diam(E) ^D if c is smaller than this constant. We apply our result to modified versions of various classical fractals. Moreover, we present a parametrised UIFS, where ω depends on c and show the inequatily H ^D (E) < diam(E) ^D , if c is small enough.     

Hausdorff measures of different dimensions are not Borel isomorphic

We show that Hausdorff measures of different dimensions are not Borel isomorphic; that is, the measure spaces (ℝ, B, H ^s ) and (ℝ, B, H ^t ) are not isomorphic if s ≠ t, s, t ∈ [0, 1], where B is the σ-algebra of Borel subsets of ℝ and H ^d is the d-dimensional Hausdorff measure. This answers a question of B. Weiss and D. Preiss. To prove our result, we apply a random construction and show that for every Borel function ƒ: ℝ → ℝ and for every d ∈ [0, 1] there exists a compact set C of Hausdorff dimension d such that ƒ(C) has Hausdorff dimension ≤ d. We also prove this statement in a more general form: If A ⊂ ℝⁿ is Borel and ƒ: A → ℝ^m is Borel measurable, then for every d ∈ [0, 1] there exists a Borel set B ⊂ A such that dim B = d·dim A and dim ƒ(B) ≤ d·dim ƒ (A). Partially supported by the Hungarian Scientific Research Fund grant no. T 49786.     

On a generalized dimension of self‐affine fractals

For a d ×d expanding matrix A, we de.ne a pseudo‐norm w (x) in terms of A and use this pseudo‐norm (instead of the Euclidean norm) to define the Hausdorff measure and the Hausdorff dimension dim^w _H E for subsets E in R ^d . We show that this new approach gives convenient estimations to the classical Hausdorff dimension dim^w _H E, and in the case that the eigenvalues of A have the same modulus, then dim^w _H E and dim_H E coincide. This setup is particularly useful to study self‐affine sets T generated by ?j (x) = A^–1(x +dj), dj ∈ R ^d , j = 1, …, N. We use it to investigate the fractality of T for the case that {?j }^N _{j =1} satisfying the open set condition as well as the cases without the open set condition. We extend some well‐known results in the self‐similar sets to the self‐af.ne sets. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An optimal extension of Marstrand’s plane-packing theorem

We prove that if F is a subset of the 2-dimensional unit sphere in $\mathbb{R}^3$, with Hausdorff dimension strictly greater than 1, and E is a subset of $\mathbb{R}^3$ such that for each $e \in F$, E contains a plane perpendicular to the vector e, then E must have positive 3-dimensional Lebesgue measure.Received: 16 April 2002     

Stability for a cap body inequality

There are inequalities of Favard and Minkowski in which equality holds for cap bodies in E ^d. By obtaining a lower bound for the first area measure, we show that if the Favard deficit is small for some body, then that body must be close, in terms of the Hausdorff metric, to a cap body. For d=3, the inequality of Minkowski is strengthened, and a weak stability result follows.Supported by National Science Foundation Grant DMS 8802674.     

Ons-sets and mutual absolute continuity of measures on homogeneous spaces

We extend a recent result of A. Jonsson about mutual absolute continuity of twoD _s -measures on ans-setF ⊂R ⁿ to the homogeneous spaces (X, d, μ) of Coifman, Weiss. Here we define Hausdorff measure, Hausdorff dimension,D _s -set andd-set relative to the measureμ. Our main result holds for so called (s, d)-sets,d ≥s, and is stronger than Jonssons result even inR ⁿ . As applications we interpret this Hausdorff dimension as a relative dimension for very regular sets and show that it in general depends strongly onμ. For this purpose we construct a strictly increasing functionf :R →R, whose measure is doubling and concentrated on a set of arbitrary small Hausdorff dimension. The extension off to a quasiconformal map of the half plane onto itself sharpens a classical example of Ahlfors-Beurling.     

Self-similar sets with optimal coverings and packings

We prove that if a self-similar set E in Rn with Hausdorff dimension s satisfies the strong separation condition, then the maximal values of the Hs-density on the class of arbitrary subsets of Rn and on the class of Euclidean balls are attained, and the inverses of these values give the exact values of the Hausdorff and spherical Hausdorff measure of E. We also show that a ball of minimal density exists, and the inverse density of this ball gives the exact packing measure of E. Lastly, we show that these elements of optimal densities allow us to construct an optimal almost covering of E by arbitrary subsets of Rn, an optimal almost covering of E by balls and an optimal packing of E.     

Arithmetic Progressions in Sets of Fractional Dimension

Let E ì \mathbbR{E \subset\mathbb{R}} be a closed set of Hausdorff dimension α. Weprove that if α is sufficiently close to 1, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then E contains non-trivial 3-term arithmetic progressions.     

On pure Goldie dimensions

In this paper, we examine the pure Goldie dimension and dual pure Goldie dimension in finitely accessible additive categories. In particular, we show that if A is an object in a finitely accessible additive category 𝒜 that has finite pure Goldie dimension n and finite dual pure Goldie dimension m, then End_𝒜(A) is semilocal and the dual Goldie dimension of End_𝒜(A) is less than or equal to n+m.     

Frequencies of digits under trigonometric perturbations

We study the Hausdorff dimension of a class of sets of real numbers defined in terms of frequencies of digits in some integer base m, with the frequencies related by trigonometric perturbations. We show in particular that the Hausdorff dimension is analytic in the parameter determining the perturbation, and we obtain estimates for the coefficients of the corresponding power series in terms of m. We also compute the first terms of the series.     

On sets of directions determined by subsets of ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

Given E ? ? ^d , d ≥ 2, define

$D(E) \equiv \left\{ {{{x - y} \over {\left| {x - y} \right|}}:x,y \in E} \right\} \subset {S^{d - 1}}$

to be the set of directions determined by E. We prove that if the Hausdorff dimension of E is greater than d ? 1, then σ(D(E)) > 0, where σ denotes the surface measure on S ^d?1. In the process, we prove some tight upper and lower bounds for the maximal function associated with the Radon-Nikodym derivative of the natural measure on D. This result is sharp, since the conclusion fails to hold if E is a (d ? 1)-dimensional hyper-plane. This result can be viewed as a continuous analog of a recent result of Pach, Pinchasi, and Sharir ([22, 23]) on directions determined by finite subsets of ? ^d . We also discuss the case when the Hausdorff dimension of E is precisely d ? 1, where some interesting counter-examples have been obtained by Simon and Solomyak ([25]) in the planar case. In response to the conjecture stated in this paper, T. Orponen and T. Sahlsten ([20]) have recently proved that if the Hausdorff dimension of E equals d ? 1 and E is rectifiable and is not contained in a hyper-pane, the Lebesgue measure of the set of directions is still positive. Finally, we show that our continuous results can be used to recover and, in some cases, improve the exponents for the corresponding results in the discrete setting for large classes of finite point sets. In particular, we prove that a finite point set P ? ? ^d , d ≥ 3, satisfying a certain discrete energy condition (Definition 3.1) determines ? #P distinct directions.

     

Linear Optimization Queries

Let Γ₀ be a set of n halfspaces in E^d (where the dimension d is fixed) and let m be a parameter, n ≤ m ≤ n^d/2. We show that Γ₀ can be preprocessed in time and space O(m^1+δ) (for any fixed δ > 0) so that given a vector c E^d and another set Γ_q of additional halfspaces, the function c · x can be optimized over the intersection of the halfspaces of Γ₀ Γ_q in time O((n/m^1/d/2 + |Γ_q|)log^4d+3n). The algorithm uses a multidimensional version of Megiddo′s parametric search technique and recent results on halfspace range reporting. Applications include an improved algorithm for computing the extreme points of an n-point set P in E^d, improved output-sensitive computation of convex hulls and Voronoi diagrams, and a Monte-Carlo algorithm for estimating the volume of a convex polyhedron given by the set of its vertices (in a fixed dimension).     

Lower Bounds for Dimensions of Sums of Sets

We study lower bounds for the Minkowski and Hausdorff dimensions of the algebraic sum E+K of two sets E,K⊂ℝ ^d .     

How fast are the particles of super-Brownian motion?

In this paper we investigate fast particles in the range and support ofsuper-Brownian motion in the historical setting. In this setting eachparticle of super-Brownian motion alive at time t is represented by apath w:[0,t]→ℝ ^d and the state of historical super-Brownian motionis a measure on the set of paths. Typical particles have Brownian paths,however in the uncountable collection of particles in the range of asuper-Brownian motion there are some which at exceptional times movefaster than Brownian motion. We determine the maximal speed of allparticles during a given time period E, which turns out to be afunction of the packing dimension of E. A path w in the support ofhistorical super-Brownian motion at time t is called a-fast if . Wecalculate the Hausdorff dimension of the set of a-fast paths in thesupport and the range of historical super-Brownian motion. A valuabletool in the proofs is a uniform dimension formula for the Browniansnake, which reduces dimension problems in the space of stopped paths to dimension problems on the line. Received: 27 January 2000 / Revised version: 28 August 2000 / Published online: 24 July 2001     

Elliptic Functions With Critical Points Eventually Mapped Onto Infinity

We consider the class of elliptic functions whose critical points in the Julia set are eventually mapped onto ∞. This paper is a continuation of our previous papers, namely [11] and [12]. We study the geometry and ergodic properties of this class of elliptic functions. In particular, we obtain a lower bound on the Hausdorff dimension of the Julia set that is bigger than the estimate proved in [11]. Let h be the Hausdorff dimension of the Julia set of f. We construct an atomless h-conformal measure m and prove the existence of a (unique up to a multiplicative constant) σ-finite f-invariant measure μ equivalent to m. The measure μ is ergodic and conservative.     

The Choquet integral representation in the affine vector-valued case

LetX be any compact convex subset of a locally convex Hausdorff space andE be a complex Banach space. We denote byA(X, E) the space of all continuous and affineE-valued functions defined onX. In this paper we prove thatX is a Choquet simplex if and only if the dual ofA(X, E) is isometrically isomorphic by a selection map toM _m (X, E*), the space ofE*-valued,w*-regular boundary measures onX. This extends and strengthens a result of G. M. Ustinov. To do this we show that for any compact convex setX, each element of the dual ofA(X, E) can be represented by a measure inM _m (X, E*) with the same norm, and this representation is unique if and only ifX is a Choquet simplex. We also prove that ifX is metrizable andE is separable then there exists a selection map from the unit ball of the dual ofA(X, E) into the unit ball ofM _m (X, E*) which is weak* to weak*-Borel measurable.This work will constitute a portion of the author's Ph.D. Thesis at the University of Illinois.     

Elliptic Functions With Critical Points Eventually Mapped Onto Infinity

We consider the class of elliptic functions whose critical points in the Julia set are eventually mapped onto ∞. This paper is a continuation of our previous papers, namely [11] and [12]. We study the geometry and ergodic properties of this class of elliptic functions. In particular, we obtain a lower bound on the Hausdorff dimension of the Julia set that is bigger than the estimate proved in [11]. Let h be the Hausdorff dimension of the Julia set of f. We construct an atomless h-conformal measure m and prove the existence of a (unique up to a multiplicative constant) σ-finite f-invariant measure μ equivalent to m. The measure μ is ergodic and conservative.     

Divergence points of self-similar measures satisfying the OSC

Recently, Barreira and Schmeling (2000) [1] and Chen and Xiong (1999) [2] have shown, that for self-similar measures satisfying the SSC the set of divergence points typically has the same Hausdorff dimension as the support K. It is natural to ask whether we obtain a similar result for self-similar measures satisfying the OSC. However, with only the OSC satisfied, we cannot do most of the work on a symbolic space and then transfer the results to the subsets of Rd, which makes things more difficult. In this paper, by the box-counting principle we show that the set of divergence points has still the same Hausdorff dimension as the support K for self-similar measures satisfying the OSC.     

Affine geometries and sets of equiorthogonal frequency hypercubes

Equiorthogonal frequency hypercubes are one particular generalization of orthogonal latin squares. A complete set of mutually equiorthogonal frequency hypercubes (MEFH) of order n and dimension d, using m distinct symbols, has (n − 1)^d/(m − 1) hypercubes. In this article, we prove that an affine geometry of dimension dh over 𝔽_m can always be used to construct a complete set of MEFH of order m^h and dimension d, using m distinct symbols. We also provide necessary and sufficient conditions for a complete set of MEFH to be equivalent to an affine geometry. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 435–441, 2000