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Kevin G. Hare

Publications and source records attributed to Kevin G. Hare.

At least 37 records · Page 2Linked to original sources
arXiv2019

The $L^q$-spectrum for a class of self-similar measures with overlap

It is known that the heuristic principle, referred to as the multifractal formalism, need not hold for self-similar measures with overlap, such as the $3$-fold convolution of the Cantor measure and certain Bernoulli convolutions. In this paper we study an important function in the multifractal theory, the $L^{q}$-spectrum, $τ(q)$, for measures of finite type, a class of self-similar measures that includes these examples. Corresponding to each measure, we introduce finitely many variants on the $% L^{q}$-spectrum which arise naturally from the finite type structure and are often easier to understand than $τ$. We show that $τ$ is always bounded by the minimum of these variants and is equal to the minimum variant for $q\geq 0$. This particular variant coincides with the $L^{q}$-spectrum of the measure $μ$ restricted to appropriate subsets of its support. If the IFS satisfies particular structural properties, which do hold for the above examples, then $τ$ is shown to be the minimum of these variants for all $q$. Under certain assumptions on the local dimensions of $μ$, we prove that the minimum variant for $q \ll 0$ coincides with the straight line having slope equal to the maximum local dimension of $μ$. Again, this is the case with the examples above. More generally, bounds are given for $τ$ and its variants in terms of notions closely related to the local dimensions of $μ$.

math.CA↗
arXiv2018

Applications of Integer and Semi-Infinite Programming to the Integer Chebyshev Problem

We consider the integer Chebyshev problem, that of minimizing the supremum norm over polynomials with integer coefficients on the interval $[0,1]$. We implement algorithms from semi-infinite programming and a branch and bound algorithm to improve on previous methods for finding integer Chebyshev polynomials of degree $n$. Using our new method, we found 16 new integer Chebyshev polynomials of degrees in the range 147 to 244.

math.NT↗
arXiv2018

The Entropy of Cantor--like measures

By a Cantor-like measure we mean the unique self-similar probability measure $μ$ satisfying $μ=\sum_{i=0}^{m-1}p_{i}μ\circ S_{i}^{-1}$ where $% S_{i}(x)=\frac{x}{d}+\frac{i}{d}\cdot \frac{d-1}{m-1}$ for integers $2\leq d 0$, $\sum p_{i}=1$. In the uniform case ($p_{i}=1/m$ for all $i$) we show how one can compute the entropy and Hausdorff dimension to arbitrary precision. In the non-uniform case we find bounds on the entropy.

math.MG↗
arXiv2018

Local dimensions of overlapping self-similar measures

We show that any equicontractive, self-similar measure arising from the IFS of contractions $(S_{j})$, with self-similar set $[0,1]$, admits an isolated point in its set of local dimensions provided the images of $S_{j}(0,1)$ (suitably) overlap and the minimal probability is associated with one (resp., both) of the endpoint contractions. Examples include $m$-fold convolution products of Bernoulli convolutions or Cantor measures with contraction factor exceeding $1/(m+1)$ in the biased case and $1/m$ in the unbiased case. We also obtain upper and lower bounds on the set of local dimensions for various Bernoulli convolutions.

math.CA↗
arXiv2018

Open maps: small and large holes with unusual properties

Let $X$ be a two-sided subshift on a finite alphabet endowed with a mixing probability measure which is positive on all cylinders in $X$. We show that there exist arbitrarily small finite overlapping union of shifted cylinders which intersect every orbit under the shift map. We also show that for any proper subshift $Y$ of $X$ there exists a finite overlapping unions of shifted cylinders such that its survivor set contains $Y$ (in particular, it can have entropy arbitrarily close to the entropy of $X$). Both results may be seen as somewhat counter-intuitive. Finally, we apply these results to a certain class of hyperbolic algebraic automorphisms of a torus.

math.DS↗
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↗
arXiv2018

On (a,b) Pairs in Random Fibonacci Sequences

We study the random Fibonacci tree, which is an infinite binary tree with non-negative integers at each node. The root consists of the number 1 with a single child, also the number 1. We define the tree recursively in the following way: if x is the parent of y, then y has two children, namely |x-y| and x+y. This tree was studied by Benoit Rittaud who proved that any pair of integers a,b that are coprime occur as a parent-child pair infinitely often. We extend his results by determining the probability that a random infinite walk in this tree contains exactly one pair (1,1), that being at the root of the tree. Also, we give tight upper and lower bounds on the number of occurrences of any specific coprime pair (a,b) at any given fixed depth in the tree.

math.NT↗
arXiv2018

The baker's map with a convex hole

We consider the baker's map $B$ on the unit square $X$ and an open convex set $H\subset X$ which we regard as a hole. The survivor set $\mathcal J(H)$ is defined as the set of all points in $X$ whose $B$-trajectories are disjoint from $H$. The main purpose of this paper is to study holes $H$ for which $\dim_H \mathcal J(H)=0$ (dimension traps) as well as those for which any periodic trajectory of $B$ intersects $\overline H$ (cycle traps). We show that any $H$ which lies in the interior of $X$ is not a dimension trap. This means that, unlike the doubling map and other one-dimensional examples, we can have $\dim_H \mathcal J(H)>0$ for $H$ whose Lebesgue measure is arbitrarily close to one. Also, we describe holes which are dimension or cycle traps, critical in the sense that if we consider a strictly convex subset, then the corresponding property in question no longer holds. We also determine $δ>0$ such that $\dim_H \mathcal J(H)>0$ for all convex $H$ whose Lebesgue measure is less than $δ$. This paper may be seen as a first extension of our work begun in [3, 4, 6, 7, 13] to higher dimensions.

math.DS↗
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.

math.DS↗
arXiv2017

A lower bound for the dimension of Bernoulli convolutions

Let $β\in(1,2)$ and let $H_β$ denote Garsia's entropy for the Bernoulli convolution $μ_β$ associated with $β$. In the present paper we show that $H_β>0.82$ for all $β\in (1, 2)$ and improve this bound for certain ranges. Combined with recent results by Hochman and Breuillard-Varjú, this yields $\dim (μ_β)\ge0.82$ for all $β\in(1,2)$. In addition, we show that if an algebraic $β$ is such that $[\mathbb{Q}(β): \mathbb{Q}(β^k)] = k$ for some $k \geq 2$, then $\dim(μ_β)=1$. Such is, for instance, any root of a Pisot number which is not a Pisot number itself.

math.DS↗
arXiv2016

On the spectra of Pisot-cyclotomic numbers

We investigate the complex spectra \[ X^{\mathcal A}(β)=\left\{\sum_{j=0}^na_jβ^j : n\in{\mathbb N},\ a_j\in{\mathcal A}\right\} \] where $β$ is a quadratic or cubic Pisot-cyclotomic number and the alphabet $\mathcal A$ is given by $0$ along with a finite collection of roots of unity. Such spectra are discrete aperiodic structures with crystallographically forbidden symmetries. We discuss in general terms under which conditions they possess the Delone property required for point sets modeling quasicrystals. We study the corresponding Voronoi tilings and we relate these structures to quasilattices arising from the cut and project method.

math-ph↗
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

Generalized Continued Logarithms and Related Continued Fractions

We study continued logarithms as introduced by Bill Gosper and studied by J. Borwein et. al.. After providing an overview of the type I and type II generalizations of binary continued logarithms introduced by Borwein et. al., we focus on a new generalization to an arbitrary integer base $b$. We show that all of our so-called type III continued logarithms converge and all rational numbers have finite type III continued logarithms. As with simple continued fractions, we show that the continued logarithm terms, for almost every real number, follow a specific distribution. We also generalize Khinchine's constant from simple continued fractions to continued logarithms, and show that these logarithmic Khinchine constants have an elementary closed form. Finally, we show that simple continued fractions are the limiting case of our continued logarithms, and briefly consider how we could generalize past continued logarithms.

math.NT↗
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.

math.DS↗
arXiv2016

Multidimensional self-affine sets: non-empty interior and the set of uniqueness

Let $M$ be a $d\times d$ contracting matrix. In this paper we consider the self-affine iterated function system $\{Mv-u, Mv+u\}$, where $u$ is a cyclic vector. Our main result is as follows: if $|\det M|\ge 2^{-1/d}$, then the attractor $A_M$ has non-empty interior. We also consider the set $\mathcal U_M$ of points in $A_M$ which have a unique address. We show that unless $M$ belongs to a very special (non-generic) class, the Hausdorff dimension of $\mathcal U_M$ is positive. For this special class the full description of $\mathcal U_M$ is given as well. This paper continues our work begun in two previous papers.

math.DS↗
arXiv2015

Two-dimensional self-affine sets with interior points, and the set of uniqueness

Let $M$ be a $2\times2$ real matrix with both eigenvalues less than~1 in modulus. Consider two self-affine contraction maps from $\mathbb R^2 \to \mathbb R^2$, \begin{equation*} T_m(v) = M v - u \ \ \mathrm{and}\ \ T_p(v) = M v + u, \end{equation*} where $u\neq0$. We are interested in the properties of the attractor of the iterated function system (IFS) generated by $T_m$ and $T_p$, i.e., the unique non-empty compact set $A$ such that $A = T_m(A) \cup T_p(A)$. Our two main results are as follows: 1. If both eigenvalues of $M$ are between $2^{-1/4}\approx 0.8409$ and $1$ in absolute value, and the IFS is non-degenerate, then $A$ has non-empty interior. 2. For almost all non-degenerate IFS, the set of points which have a unique address is of positive Hausdorff dimension -- with the exceptional cases fully described as well. This paper continues our work begun in [11].

math.DS↗
arXiv2015

On a family of self-affine sets: topology, uniqueness, simultaneous expansions

Let $β_1,β_2>1$ and $T_i(x,y) = \bigl(\frac{x+i}{β_1}, \frac{y+i}{β_2}\bigr),\ i\in\{\pm1\}$. Let $A := A_{β_1, β_2}$ be the unique compact set satisfying $A = T_{1}(A) \cup T_{-1}(A)$. In this paper we give a detailed analysis of $A$, and the parameters $(β_1, β_2)$ where$A$ satisfies various topological properties. In particular, we show that if $β_1<β_2<1.202$,then $A$ has a non-empty interior, thus significantly improving the bound from [1]. In the opposite direction,we prove that the connectedness locus for this family studied in [16] is not simply connected.We prove that the set of points of $A$ which have a unique address has positive Hausdorff dimension for all $(β_1,β_2)$.Finally, we investigate simultaneous $(β_1,β_2)$-expansions of reals, which were the initial motivation for studying this family in [5].

math.DS↗