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Kevin G. Hare

Publications and source records attributed to Kevin G. Hare.

At least 19 recordsLinked to original sources
arXiv2026

Non-Expansive Matrix Based number Systems

An integer $d$-dimensional vector is usually represented by $d$ strings, which express the value of each component separately. This article is devoted to number systems that allow representing integer vectors using a single string of digits. This system is given by an integer square matrix $M$ and a finite set $\mathcal{D}$ of symbols, called digits. We focus on full number systems, in which every vector $\vec{x} \in \mathbb{Z}^n$ can be written in the form $\vec{x}=\sum_{i=0}^{k-1} M^i d_i$ with $d_i \in \mathcal{D}$ and thus represented by the string $d_{k-1}d_{k-2}\cdots d_1d_0$. In contrast to the very well-described case where the base $M$ is expansive, we focus on systems in which $M$ is similar to a Jordan block with the eigenvalue $\pm 1$. We follow the work of Caldwell, Hare, and Vávra who initiated the study of such systems. First, we address the question of choosing a minimal sized digit set $\mathcal{D}$. Then we describe optimal representations for vectors from $\mathbb{Z}^2$, both with respect to their length and the occurrence of non-zero digits.

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arXiv2026

Intermediate Assouad-like dimensions for measures

The upper and lower Assouad dimensions of a metric space are local variants of the box dimensions of the space and provide quantitative information about the `thickest' and `thinnest' parts of the set. Less extreme versions of these dimensions for sets have been introduced, including the upper and lower quasi-Assouad dimensions, $θ$-Assouad spectrum, and $Φ$-dimensions. In this paper, we study the analogue of the upper and lower $Φ$-dimensions for measures. We give general properties of such dimensions, as well as more specific results for self-similar measures satisfying various separation properties and discrete measures.

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arXiv2026

When the weak separation condition implies the generalize finite type in $\mathbb{R}^d$

Let $\mathcal{S}$ be an iterated function system in $\mathbb{R}^d$, with full support and some restrictions on the allowable rotations. We show that $\mathcal{S}$ satisfies the weak separation condition if and only if it satisfies the generalized finite-type condition. With this in mind, we extend the notion of net intervals from $\mathbb{R}$ to $\mathbb{R}^d$. We also use net intervals to calculate the local dimension of a self-similar measure with the finite-type condition and full support.

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arXiv2025

Greedy Beta-expansions for families of Salem numbers

We give criteria for finding the greedy $β$-expansion for $1$ for families of Salem numbers that approach a given Pisot number. We show that these expansions are related to the greedy expansion under the Pisot base. This expands on the work of Hare and Tweedle.

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arXiv2025

Bounding the Local Dimension of the Convolution of Measures

We study the local dimension of the convolution of two measures. We give conditions for bounding the local dimension of the convolution on the basis of the local dimension of one of them. Moreover, we give a formula for the local dimension of some special points in the support of the convolution.

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arXiv2024

Totally real algebraic numbers in generalized Mandelbrot set

In this article, we study some potential theoretical and topological aspects of the generalized Mandelbrot set introduced by Baker and DeMarco. For $α$ real, we study the set of all totally real algebraic parameters $c$ such that $α$ is preperiodic under the iteration of the one-parameter family $f_c(x) = x^2 + c$. We show that when $|α| < 2$ and rational then the set of totally real algebraic parameters $c$ with this property is finite, whereas if $|α| \geq 2$ and rational then this set is countably infinite. As an unexpected consequence of this study, we also show that when $|α| \geq 2$ then parameters $c$ such that $α$ is $f_c$-periodic are necessarily real. As a special case, we classify all totally real algebraic integers $c$ such that $α= \pm1$ is preperiodic.

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arXiv2024

Equicontractive weak separation property on the line does not imply convex finite type condition

Let $\{S_1, S_2, \dots, S_n\}$ be an iterated function system on $\mathbb{R}$ with attractor $K$. It is known that if the iterated function system satisfies the weak separation property and $K = [0,1]$ then the iterated function system also satisfies the convex finite type condition. We show that the condition $K = [0,1]$ is necessary. That is, we give two examples of iterated function systems on $\mathbb{R}$ satisfying weak separation condition, and $0< \dim_H(K) < 1$ such that the IFS does not satisfy the convex finite type condition.

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arXiv2023

Computing Garsia Entropy for Bernoulli Convolutions with Algebraic Parameters

We introduce a parameter space containing all algebraic integers $β\in(1,2]$ that are not Pisot or Salem numbers, and a sequence of increasing piecewise continuous function on this parameter space which gives a lower bound for the Garsia entropy of the Bernoulli convolution $ν_β$. This allows us to show that $\mathrm{dim}_\mathrm{H} (ν_β)=1$ for all $β$ with representations in certain open regions of the parameter space.

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arXiv2023

Self-similar sets and self-similar measures in the $p$-adics

In this paper we investigate $p$-adic self-similar sets and $p$-adic self-similar measures. We show that $p$-adic self-similar sets are $p$-adic path set fractals, and that the converse is not necessarily true. For $p$-adic self-similar sets and $p$-adic self-similar measures, we show the existence of a unique essential class. We show that, under mild assumptions, the decimation of $p$-adic self-similar sets is maximal. For $p$-adic self-similar measures, we show that many results involving local dimension are similar to those of their real counterparts, with fewer complications. Most of these results use the additional structure of self-similarity, and are not true in general for $p$-adic path set fractals.

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arXiv2023

Computational progress on the unfair 0-1 polynomial Conjecture

Let $c(x)$ be a monic integer polynomial with coefficients $0$ or $1$. Write $c(x) = a(x) b(x)$ where $a(x)$ and $b(x)$ are monic polynomials with non-negative real (not necessarily integer) coefficients. The unfair 0--1 polynomial conjecture states that $a(x)$ and $b(x)$ are necessarily integer polynomials with coefficients $0$ or $1$. Let $a(x)$ be a candidate factor of a (currently unknown) 0--1 polynomial. We will assume that we know if a coefficient is $0$, $1$ or strictly between $0$ and $1$, but that we do not know the precise value of non-integer coefficients. Given this candidate $a(x)$, this paper gives an algorithm to either find a $b(x)$ and $c(x)$ with $a(x) b(x) = c(x)$ such that $b(x)$ has non-negative real coefficients and $c(x)$ has coefficients $0$ or $1$, or (often) shows that no such $c(x)$ and $b(x)$ exist. Using this algorithm, we consider all candidate factors with degree less than or equal to 15. With the exception of 975 candidate factors (out of a possible 7141686 cases), this algorithm shows that there do not exist $b(x)$ with non-negative real coefficients and $c(x)$ with coefficients $0$ or $1$ such that $a(x) b(x) = c(x)$.

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arXiv2022

The Minkowski sum of linear Cantor sets

Let $C$ be the classical middle third Cantor set. It is well known that $C+C = [0,2]$ (Steinhaus, 1917). (Here $+$ denotes the Minkowski sum.) Let $U$ be the set of $z \in [0,2]$ which have a unique representation as $z = x + y$ with $x, y \in C$ (the set of uniqueness). It isn't difficult to show that $\dim_H U = \log(2) / \log(3)$ and $U$ essentially looks like $2C$. Assuming $0,n-1 \in A \subset \{0,1,\dots,n-1\}$, define $C_A = C_{A,n}$ as the linear Cantor set which the attractor of the iterated function system \[ \{ x \mapsto (x + a) / n: a \in A \}. \] We consider various properties of such linear Cantor sets. Our main focus will be on the structure of $C_{A,n}+C_{A,n}$ depending on $n$ and $A$ as well as the properties of the set of uniqueness $U_A$.

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arXiv2022

Self-similar measures with unusual local dimension properties

Let $μ$ be a self-similar measure satisfying the finite type condition. It is known that the set of attainable local dimensions for such a measure is a union of disjoint intervals, where some intervals may be degenerate points. Despite this, it has not been shown if this full complexity of attainable local dimensions is achievable. In this paper we give two different constructions. The first is a measure $μ$ where the set of all attainable local dimensions is the union of an interval union and an arbitrary number of disjoint points. The second is a measure $μ$ where the set of all attainable local dimensions is the union of an arbitrary number of disjoint intervals. As an application to these construction, we study the multi-fractal spectrum $f_μ(α)$ and the $L^q$-spectrum $τ_μ(q)$ of these measures. We given an example of a $μ$ where $f_μ(α)$ is not concave, and where $τ_μ(q)$ has two points of non-differentiability.

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arXiv2021

Non-expansive matrix number systems with bases similar to $J_n(1)$

We study representations of integral vectors in a number system with a matrix base $M$ and vector digits. We focus on the case when $M$ is similar to $J_n$, the Jordan block of $1$ of size $n$. If $M=J_2$, we classify digit sets of size 2 allowing representation of the whole $\mathbb{Z}^2$. For $J_n$ with $n\geq 3$, it is shown that three digits suffice to represent all of $\mathbb{Z}^n$. For bases similar to $J_n$, at most $n$ digits are required, with the exception of $n=1$. Moreover, the language of strings representing the zero vector with $M=J_2$ and the digits $(0,\pm 1)^T$ is shown not to be context-free, but to be recognizable by a Turing machine with logarithmic memory.

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arXiv2021

On a family of Self-Affine IFS whose attractors have a non-fractal top

Let $0< λ< μ<1$ and $λ+μ>1$. In this note we prove that for the vast majority of such parameters the top of the attractor $A_{λ,μ}$ of the IFS $\{(λx,μy), (μx+1-μ, λy+1-λ)\}$ is the graph of a continuous, strictly increasing function. Despite this, for most parameters, $A_{λ, μ}$ has a box dimension strictly greater than 1, showing that the upper boundary is not representative of the complexity of the fractal. Finally, we prove that if $λμ\ge 2^{-1/6}$, then $A_{λ,μ}$ has a non-empty interior.

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arXiv2020

Conjugates of Pisot numbers

In this paper we investigate the Galois conjugates of a Pisot number $q \in (m, m+1)$, $m \geq 1$. In particular, we conjecture that for $q \in (1,2)$ we have $|q'| \geq \frac{\sqrt{5}-1}{2}$ for all conjugates $q'$ of $q$. Further, for $m \geq 3$, we conjecture that for all Pisot numbers $q \in (m, m+1)$ we have $|q'| \geq \frac{m+1-\sqrt{m^2+2m-3}}{2}$. A similar conjecture if made for $m =2$. We conjecture that all of these bounds are tight. We provide partial supporting evidence for this conjecture. This evidence is both of a theoretical and computational nature. Lastly, we connect this conjecture to a result on the dimension of Bernoulli convolutions parameterized by $β$, whose conjugate is the reciprocal of a Pisot number.

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arXiv2019

On Newman and Littlewood polynomials with prescribed number of zeros inside the unit disk

We study $\{0, 1\}$ and $\{-1, 1\}$ polynomials $f(z)$, called Newman and Littlewood polynomials, that have a prescribed number $N(f)$ of zeros in the open unit disk $\mathcal{D} = \{z \in \mathbb{C}: |z| < 1\}$. For every pair $(k, n) \in \mathbb{N}^2$, where $n \geq 7$ and $k \in [3, n-3]$, we prove that it is possible to find a $\{0, 1\}$--polynomial $f(z)$ of degree $\text{deg }{f}=n$ with non--zero constant term $f(0) \ne 0$, such that $N(f)=k$ and $f(z) \ne 0$ on the unit circle $\partial\mathcal{D}$. On the way to this goal, we answer a question of D.~W.~Boyd from 1986 on the smallest degree Newman polynomial that satisfies $|f(z)| > 2$ on the unit circle $\partial \mathcal{D}$. This polynomial is of degree $38$ and we use this special polynomial in our constructions. We also identify (without a proof) all exceptional $(k, n)$ with $k \in \{1, 2, 3, n-3, n-2, n-1\}$, for which no such $\{0, 1\}$--polynomial of degree $n$ exists: such pairs are related to regular (real and complex) Pisot numbers. Similar, but less complete results for $\{-1, 1\}$ polynomials are established. We also look at the products of spaced Newman polynomials and consider the rotated large Littlewood polynomials. Lastly, based on our data, we formulate a natural conjecture about the statistical distribution of $N(f)$ in the set of Newman and Littlewood polynomials.

math.NT↗
arXiv2019

Local dimensions of measures of finite type III -- Measures that are not equicontractive

We extend the study of the multifractal analysis of the class of equicontractive self-similar measures of finite type to the non-equicontractive setting. Although stronger than the weak separation condition, the finite type property includes examples of IFS that fail the open set condition. The important combinatorial properties of equicontractive self-similar measures of finite type are extended to the non-equicontractive setting and we prove that many of the results from the equicontractive case carry over to this new, more general, setting. In particular, previously it was shown that if an equicontractive self-similar measure of finite type was {\em regular}, then the calculations of local dimensions were relatively easy. We modify this definition of regular to define measures to be {\em generalized regular}. This new definition will include the non-equicontractive case and obtain similar results. Examples are studied of non-equicontractive self-similar generalized regular measures, as well as equicontractive self-similar measures which generalized regular in this new sense, but which are not regular.

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