arXiv subjects

Kathryn E. Hare

Publications and source records attributed to Kathryn E. Hare.

30 records · Page 2Linked to original sources

arXiv2018

The Assouad spectrum and the quasi-Assouad dimension: a tale of two spectra

We consider the Assouad spectrum, introduced by Fraser and Yu, along with a natural variant that we call the `upper Assouad spectrum'. These spectra are designed to interpolate between the upper box-counting and Assouad dimensions. It is known that the Assouad spectrum approaches the upper box-counting dimension at the left hand side of its domain, but does not necessarily approach the Assouad dimension on the right. Here we show that it necessarily approaches the \emph{quasi-Assouad dimension} at the right hand side of its domain. We further show that the upper Assouad spectrum can be expressed in terms of the Assouad spectrum, thus motivating the definition used by Fraser-Yu. We also provide a large family of examples demonstrating new phenomena relating to the form of the Assouad spectrum. For example, we prove that it can be strictly concave, exhibit phase transitions of any order, and need not be piecewise differentiable.

math.CA↗

arXiv2017

Local dimensions of random homogeneous self-similar measures: strong separation and finite type

We study the multifractal analysis of self-similar measures arising from random homogeneous iterated function systems. Under the assumption of the uniform strong separation condition, we see that this analysis parallels that of the deterministic case. The overlapping case is more complicated; we introduce the notion of finite type for random homogeneous iterated function systems and give a formula for the local dimensions of finite type, regular, random homogeneous self-similar measures in terms of Lyapunov exponents of certain transition matrices. We show that almost all points with respect to this measure are described by a distinguished subset called the essential class, and that the dimension of the support can be computed almost surely from knowledge of this essential class. For a special subcase, that we call commuting, we prove that the set of attainable local dimensions is almost surely a closed interval. Particular examples of such random measures are analyzed in more detail.

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arXiv2017

Transferring spherical multipliers on compact symmetric spaces

We prove a two-sided transference theorem between $L^{p}$ spherical multipliers on the compact symmetric space $U/K$ and $L^{p}$ multipliers on the vector space $i\mathfrak{p},$ where the Lie algebra of $U$ has Cartan decomposition $\mathfrak{k\oplus }i\mathfrak{p}$. This generalizes the classic theorem transference theorem of deLeeuw relating multipliers on $% L^{p}(\mathbb{T)}$ and $L^{p}(\mathbb{R)}$.

math.FA↗

arXiv2016

Local dimensions of measures of finite type on the torus

The structure of the set of local dimensions of a self-similar measure has been studied by numerous mathematicians, initially for measures that satisfy the open set condition and, more recently, for measures on $\mathbb{R}$ that are of finite type. In this paper, our focus is on finite type measures defined on the torus, the quotient space $\mathbb{R}\backslash \mathbb{Z}$. We give criteria which ensures that the set of local dimensions of the measure taken over points in special classes generates an interval. We construct a non-trivial example of a measure on the torus that admits an isolated point in its set of local dimensions. We prove that the set of local dimensions for a finite type measure that is the quotient of a self-similar measure satisfying the strict separation condition is an interval. We show that sufficiently many convolutions of Cantor-like measures on the torus never admit an isolated point in their set of local dimensions, in stark contrast to such measures on $\mathbb{R}$. Further, we give a family of Cantor-like measures on the torus where the set of local dimensions is a strict subset of the set of local dimensions, excluding the isolated point, of the corresponding measures on $\mathbb{R}$.

math.DS↗

arXiv2016

Local dimensions of measures of finite type II - Measures without full support and with non-regular probabilities

Consider a sequence of linear contractions $S_{j}(x)=\varrho x+d_{j}$ and probabilities $p_{j}>0$ with $\sum p_{j}=1$. We are interested in the self-similar measure $μ=\sum p_{j}μ\circ S_{j}^{-1}$, of finite type. In this paper we study the multi-fractal analysis of such measures, extending the theory to measures arising from non-regular probabilities and whose support is not necessarily an interval. Under some mild technical assumptions, we prove that there exists a subset of supp$μ$ of full $μ$ and Hausdorff measure, called the truly essential class, for which the set of (upper or lower) local dimensions is a closed interval. Within the truly essential class we show that there exists a point with local dimension exactly equal to the dimension of the support. We give an example where the set of local dimensions is a two element set, with all the elements of the truly essential class giving the same local dimension. We give general criteria for these measures to be absolutely continuous with respect to the associated Hausdorff measure of their support and we show that the dimension of the support can be computed using only information about the essential class. To conclude, we present a detailed study of three examples. First, we show that the set of local dimensions of the biased Bernoulli convolution with contraction ratio the inverse of a simple Pisot number always admits an isolated point. We give a precise description of the essential class of a generalized Cantor set of finite type. Lastly, we study a maximal loop class that is not truly essential.

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arXiv2016

The Ubiquity of Sidon Sets That Are Not $I_0$

We prove that every infinite, discrete abelian group admits a pair of $I_0$ sets whose union is not $I_0$. In particular, this implies that every such group contains a Sidon set that is not $I_{0}$.

math.CA↗

arXiv2015

The absolute continuity of convolutions of orbital measures in symmetric spaces

We characterize the absolute continuity of convolution products of orbital measures on the classical, irreducible Riemannian symmetric spaces $G/K$ of Cartan type $III$, where $G$ is a non-compact, connected Lie group and $K$ is a compact, connected subgroup. By the orbital measures, we mean the uniform measures supported on the double cosets, $KzK,$ in $G$. The characterization can be expressed in terms of dimensions of eigenspaces or combinatorial properties of the annihilating roots of the elements $z$. A consequence of our work is to show that the convolution product of any rank% $G/K,$ continuous, $K$-bi-invariant measures is absolutely continuous in any of these symmetric spaces, other than those whose restricted root system is type $A_{n}$ or $D_{3}$, when rank$G/K$ $+1$ is needed.

math.RT↗

arXiv2015

Local dimensions of measures of finite type

We study the multifractal analysis of a class of equicontractive, self-similar measures of finite type, whose support is an interval. Finite type is a property weaker than the open set condition, but stronger than the weak open set condition. Examples include Bernoulli convolutions with contraction factor the inverse of a Pisot number and self-similar measures associated with $m$-fold sums of Cantor sets with ratio of dissection $1/R$ for integer $R\leq m$. We introduce a combinatorial notion called a loop class and prove that the set of attainable local dimensions of the measure at points in a positive loop class is a closed interval. We prove that the local dimensions at the periodic points in the loop class are dense and give a simple formula for those local dimensions. These self-similar measures have a distinguished positive loop class called the essential class. The set of points in the essential class has full Lebesgue measure in the support of the measure and is often all but the two endpoints of the support. Thus many, but not all, measures of finite type have at most one isolated point in their set of local dimensions. We give examples of Bernoulli convolutions whose sets of attainable local dimensions consist of an interval together with an isolated point. As well, we give an example of a measure of finite type that has exactly two distinct local dimensions.

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arXiv2015

Exact Kronecker Constants of Three Element Sets

For any three element set of positive integers, $\{a,b,n\}$, with $a<b<n$, $n$ sufficiently large and $\gcd(a,b)=1$, we find the least $α$ such that given any real numbers $t_1$, $t_2$, $t_3$, there is a real number $x$ such that \begin{equation*} \max \{\left\langle ax-t_{1}\right\rangle ,\left\langle bx-t_{2}\right\rangle ,\left\langle nx-t_{3}\right\rangle \}\leq α, \end{equation*} where $\left\langle \cdot \right\rangle $ denotes the distance to the nearest integer. The number $α$ is known as the angular Kronecker constant of $\{a,b,n\}$. We also find the least $β$ such that the same inequality holds with upper bound $β$ when we consider only approximating $t_{1},t_{2},t_{3}$ $\in \{0,1/2\}$, the so-called binary Kronecker constant. The answers are complicated and depend on the congruence of $n\mod(a+b)$. Surprisingly, the angular and binary Kronecker constants agree except if $n\equiv a^{2}\mod(a+b)$.

math.CA↗

arXiv2014

Characterizing the absolute continuity of the convolution of orbital measures in a classical Lie algebra

Let $\mathfrak{g}$ be a compact, simple Lie algebra of dimension $d$. It is a classical result that the convolution of any $d$ non-trivial, $G$ -invariant, orbital measures is absolutely continuous with respect to Lebesgue measure on $\mathfrak{g}$ and the sum of any $d$ non-trivial orbits has non-empty interior. The number $d$ was later reduced to the rank of the Lie algebra (or rank $+1$ in the case of type $A_{n}$). More recently, the minimal integer $k=k(X)$ such that the $k$-fold convolution of the orbital measure supported on the orbit generated by $X$ is an absolutely continuous measure was calculated for each $X\in \mathfrak{g}$. In this paper $\mathfrak{g}$ is any of the classical, compact, simple Lie algebras. We characterize the tuples $(X_{1},...,X_{L})$, with $X_{i}\in \mathfrak{g},$ which have the property that the convolution of the $L$ -orbital measures supported on the orbits generated by the $X_{i}$ is absolutely continuous and, equivalently, the sum of their orbits has non-empty interior. The characterization depends on the Lie type of $ \mathfrak{g}$ and the structure of the annihilating roots of the $X_{i}$. Such a characterization was previously known only for type $A_{n}$.

math.FA↗

arXiv2011

Upper and Lower Bounds for Kronecker Constants of Three-Element Sets of Integers

Various upper and lower bounds are provided for the (angular) Kronecker constants of sets of integers. Some examples are provided where the bounds are attained. It is proved that 5=16 bounds the angular Kronecker constants of 3-element sets of positive integers. However, numerous examples suggest that the minimum upper bound is 1=4 for 3-element sets of positive integers.

math.FA↗

arXiv2010

Classifying Cantor Sets by their Fractal Dimensions

In this article we study Cantor sets defined by monotone sequences, in the sense of Besicovitch and Taylor. We classify these Cantor sets in terms of their h-Hausdorff and h-Packing measures, for the family of dimension functions h, and characterize this classification in terms of the underlying sequences.

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