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Kevin Hare

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Papers by Kevin Hare

On cycles for the doubling map which are disjoint from an interval

Let T : [0, 1] → [0, 1] be the doubling map and let 0 < a < b < 1. We say that an integer n ≥ 3 i... more Let T : [0, 1] → [0, 1] be the doubling map and let 0 < a < b < 1. We say that an integer n ≥ 3 is bad for (a, b) if all n-cycles for T intersect (a, b). Let B(a, b) denote the set of all n which are bad for (a, b). In this paper we completely describe the sets: D 2 = {(a, b) : B(a, b) is finite} and D 3 = {(a, b) : B(a, b) = ∅}. In particular, we show that if ba < 1 6 , then (a, b) ∈ D 2 , and if ba ≤ 2 15 , then (a, b) ∈ D 3 , both constants being sharp.

Growth Degree Classification for Finitely Generated Semigroups of Integer Matrices

Let A be a finite set of d × d matrices with integer entries and let mn(A) be the maximum norm of... more Let A be a finite set of d × d matrices with integer entries and let mn(A) be the maximum norm of a product of n elements of A. In this paper, we classify gaps in the growth of mn(A); specifically, we prove that limn→∞ log mn(A)/ log n ∈ Z 0 ∪ {∞}. This has applications to the growth of regular sequences as defined by Allouche and Shallit.

Stolarsky’s conjecture and the sum of digits of polynomial values

Let sq(n) denote the sum of the digits in the q-ary ex-pansion of an integer n. In 1978, Stolarsk... more

An explicit counterexample to the Lagarias-Wang finiteness conjecture

The joint spectral radius of a finite set of real d × d matrices is defined to be the maximum pos... more The joint spectral radius of a finite set of real d × d matrices is defined to be the maximum possible exponential rate of growth of long products of matrices drawn from that set. A set of matrices is said to have the finiteness property if there exists a periodic product which achieves this maximal rate of growth. J. C. Lagarias and Y. Wang conjectured in 1995 that every finite set of real d × d matrices satisfies the finiteness property. However, T. Bousch and J. Mairesse proved in 2002 that counterexamples to the finiteness conjecture exist, showing in particular that there exists a family of pairs of 2 × 2 matrices which contains a counterexample. Similar results were subsequently given by V. D. Blondel, J. Theys and A. A. Vladimirov and by V. S. Kozyakin, but no explicit counterexample to the finiteness conjecture has so far been given. The purpose of this paper is to resolve this issue by giving the first completely explicit description of a counterexample to the Lagarias-Wang finiteness conjecture. Namely, for the set Aα * := 1 1 0 1 , α * 1 0 1 1 we give an explicit value of α * ≃ 0.749326546330367557943961948091344672091327370236064317358024. .. such that Aα * does not satisfy the finiteness property.

THE SUM OF DIGITS OF n AND n²

Let sq(n) denote the sum of the digits in the q-ary expansion of an integer n. In 2005, Melfi exa... more Let sq(n) denote the sum of the digits in the q-ary expansion of an integer n. In 2005, Melfi examined the structure of n such that s2(n) = s2(n²). We extend this study to the more general case of generic q and polynomials p(n), and obtain, in particular, a refinement of Melfi’s result. We also give a more detailed analysis of the special case p(n) = n², looking at the subsets of n where sq(n) = sq(n²) = k for fixed k.

There Are No Two Non-Real Conjugates of a Pisot Number with the Same Imaginary

We show that the number α = (1+ 3 + 2 √ 5)/2 with minimal polynomial x 4 −2x 3 +x−1 is the only P... more We show that the number α = (1+ 3 + 2 √ 5)/2 with minimal polynomial x 4 −2x 3 +x−1 is the only Pisot number whose four distinct conjugates α 1 , α 2 , α 3 , α 4 satisfy the additive relation α 1 + α 2 = α 3 + α 4. This implies that there exists no two non-real conjugates of a Pisot number with the same imaginary part and also that at most two conjugates of a Pisot number can have the same real part. On the other hand, we prove that similar four term equations α 1 = α 2 + α 3 + α 4 or α 1 + α 2 + α 3 + α 4 = 0 cannot be solved in conjugates of a Pisot number α. We also show that the roots of the Siegel's polynomial x 3 − x − 1 are the only solutions to the three term equation α 1 + α 2 + α 3 = 0 in conjugates of a Pisot number. Finally, we prove that there exists no Pisot number whose conjugates satisfy the relation α 1 = α 2 + α 3 .

Random Fibonacci sequences from balancing words

arXiv: Number Theory, 2019

We study growth rates of random Fibonacci sequences of a particular structure. A random Fibonacci... more We study growth rates of random Fibonacci sequences of a particular structure. A random Fibonacci sequence is an integer sequence starting with $1,1$ where the next term is determined to be either the sum or the difference of the two preceding terms where the choice of taking either the sum or the difference is chosen randomly at each step. In 2012, McLellan proved that if the pluses and minuses follow a periodic pattern and $G_n$ is the $n$th term of the resulting random Fibonacci sequence, then \begin{equation*} \lim_{n\rightarrow\infty}|G_n|^{1/n} \end{equation*} exists. We extend her results to recurrences of the form $G_{m+2} = \alpha_m G_{m+1} \pm G_{m}$ if the choices of pluses and minuses, and of the $\alpha_m$ follow a balancing word type pattern.

Further Results on Derived Sequences

In 2003 Cohen and Iannucci introduced a multiplicative arithmetic function D by assigning D(p a )... more In 2003 Cohen and Iannucci introduced a multiplicative arithmetic function D by assigning D(p a ) = ap ai1 when p is a prime and a is a positive integer. They deflned D 0 (n) = n and D k (n) = D(D ki1 (n)) and they called fD k (n)g 1=0 the derived sequence of n. This paper answers some open questions about the function D and its iterates. We show how to construct derived sequences of arbitrary cycle size, and we give examples for cycles of lengths up to 10. Given n, we give a method for computing m such that D(m) = n, up to a square free unitary factor.

Generalized Continued Logarithms and Related Continued Fractions

J. Integer Seq., 2017

We study continued logarithms as introduced by Bill Gosper and studied by J. Borwein et. al.. Aft... more 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.

Three Series for the Generalized Golden Mean

arXiv: Number Theory, 2014

As is well-known, the ratio of adjacent Fibonacci numbers tends to � = (1+ √ 5)/2, and the ratio ... more

There are no two non-real conjugates of a Pisot number with the same imaginary part

arXiv: Number Theory, 2014

We show that the number $\alpha=(1+\sqrt{3+2\sqrt{5}})/2$ with minimal polynomial $x^4-2x^3+x-1$ ... more We show that the number $\alpha=(1+\sqrt{3+2\sqrt{5}})/2$ with minimal polynomial $x^4-2x^3+x-1$ is the only Pisot number whose four distinct conjugates $\alpha_1,\alpha_2,\alpha_3,\alpha_4$ satisfy the additive relation $\alpha_1+\alpha_2=\alpha_3+\alpha_4$. This implies that there exists no two non-real conjugates of a Pisot number with the same imaginary part and also that at most two conjugates of a Pisot number can have the same real part. On the other hand, we prove that similar four term equations $\alpha_1 = \alpha_2 + \alpha_3+\alpha_4$ or $\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4 =0$ cannot be solved in conjugates of a Pisot number $\alpha$. We also show that the roots of the Siegel's polynomial $x^3-x-1$ are the only solutions to the three term equation $\alpha_1+\alpha_2+\alpha_3=0$ in conjugates of a Pisot number. Finally, we prove that there exists no Pisot number whose conjugates satisfy the relation $\alpha_1=\alpha_2+\alpha_3$.

de Théorie des Nombres de Bordeaux 00 ( XXXX ) , 000 – 000 Patterns and Periodicity in a Family of Resultants par

Given a monic degree N polynomial f(x) ∈ Z[x] and a non-negative integer l, we may form a new mon... more Given a monic degree N polynomial f(x) ∈ Z[x] and a non-negative integer l, we may form a new monic degree N polynomial fl(x) ∈ Z[x] by raising each root of f to the lth power. We generalize a lemma of Dobrowolski to show that if m < n and p is prime then p divides the resultant of fpm and fpn . We then consider the function (j, k) 7→ Res(fj , fk) mod p . We show that for fixed p and m that this function is periodic in both j and k, and exhibits high levels of symmetry. Some discussion of its structure as a union of lattices is also given.

Conjugates of Pisot numbers

In this paper we investigate the Galois conjugates of a Pisot number $q \in (m, m+1)$, $m \geq 1$... more 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 $\beta$, whose conjugate is the reciprocal of a Pisot number.

Some Properties of Even Moments of Uniform Random Walks

We build upon previous work on the densities of uniform random walks in higher dimensions, explor... more

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

We study $\{0, 1\}$ and $\{-1, 1\}$ polynomials $f(z)$, called Newman and Littlewood polynomials,... more 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 $\mathcal{D} = \{z \in \mathbb{C}: |z| 2$ 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.

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 d... more 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 Jn, the Jordan block of 1 of size n. If M = J2, we classify digit sets of size 2 allowing representation of the whole Z. For Jn with n ≥ 3, it is shown that three digits suffice to represent all of Z. For bases similar to Jn, at most n digits are required, with the exception of n = 1. Moreover, the language of strings representing the zero vector with M = J2 and the digits (0,±1) is shown not to be context-free, but to be recognizable by a Turing machine with logarithmic memory.

Computing Garsia entropy for Bernoulli convolutions with algebraic parameters *

Nonlinearity

We introduce a parameter space containing all algebraic integers β ∈ (1, 2] that are not Pisot or... more We introduce a parameter space containing all algebraic integers β ∈ (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 dimH(ν β ) = 1 for all β with representations in certain open regions of the parameter space.

On (a,b) pairs in random Fibonacci sequences

Journal of Number Theory

We study the random Fibonacci tree, which is an infinite binary tree with nonnegative integers at... more We study the random Fibonacci tree, which is an infinite binary tree with nonnegative 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 [6] 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.

Most Reinhardt polygons are sporadic

Geometriae Dedicata

A Reinhardt polygon is a convex n-gon that, for n not a power of 2, is optimal in three different... more A Reinhardt polygon is a convex n-gon that, for n not a power of 2, is optimal in three different geometric optimization problems, for example, it has maximal perimeter relative to its diameter. Some such polygons exhibit a particular periodic structure; others are termed sporadic. Prior work has described the periodic case completely, and has shown that sporadic Reinhardt polygons occur for all n of the form n = pqr with p and q distinct odd primes and r ≥ 2. We show that (dihedral equivalence classes of) sporadic Reinhardt polygons outnumber the periodic ones for almost all n, and find that this first occurs at n = 105. We also determine a formula for the number of sporadic Reinhardt polygons when n = 2pq with p and q distinct odd primes.

A Lower Bound for the Dimension of Bernoulli Convolutions

Experimental Mathematics

Let β ∈ (1, 2) and let H β denote Garsia's entropy for the Bernoulli convolution µ β associated w... more Let β ∈ (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 ∈ (1, 2) and improve this bound for certain ranges. Combined with recent results by Hochman and Breuillard-Varjú, this yields dim(µ β) ≥ 0.82 for all β ∈ (1, 2). In addition, we show that if an algebraic β is such that [Q(β) : Q(β k)] = k for some k ≥ 2 then dim(µ β) = 1. Such is, for instance, any root of a Pisot number which is not a Pisot number itself.