arXiv subjects

Steven J. Miller

Publications and source records attributed to Steven J. Miller.

At least 109 records · Page 6Linked to original sources

arXiv2022

Distinct Angles in General Position

The Erdős distinct distance problem is a ubiquitous problem in discrete geometry. Somewhat less well known is Erdős' distinct angle problem, the problem of finding the minimum number of distinct angles between $n$ non-collinear points in the plane. Recent work has introduced bounds on a wide array of variants of this problem, inspired by similar variants in the distance setting. In this short note, we improve the best known upper bound for the minimum number of distinct angles formed by $n$ points in general position from $O(n^{\log_2(7)})$ to $O(n^2)$. Before this work, similar bounds relied on projections onto a generic plane from higher dimensional space. In this paper, we employ the geometric properties of a logarithmic spiral, sidestepping the need for a projection. We also apply this configuration to reduce the upper bound on the largest integer such that any set of $n$ points in general position has a subset of that size with all distinct angles. This bound is decreased from $O(n^{\log_2(7)/3})$ to $O(n^{1/2})$.

cs.CG↗

arXiv2022

Class Numbers and Pell's Equation $x^2 + 105y^2 = z^2$

Two well-studied Diophantine equations are those of Pythagorean triples and elliptic curves, for the first we have a parametrization through rational points on the unit circle, and for the second we have a structure theorem for the group of rational solutions. Recently, Yekutieli discussed a connection between these two problems, and described the group structure of Pythagorean triples and the number of triples for a given hypotenuse. In arXiv:2112.03663 we generalized these methods and results to Pell's equation. We find a similar group structure and count on the number of solutions for a given $z$ to $x^2 + Dy^2 = z^2$ when $D$ is 1 or 2 modulo 4 and the class group of $\mathbb{Q}[\sqrt{-D}]$ is a free $\mathbb{Z}_2$ module, which always happens if the class number is at most 2. In this paper, we discuss the main results of arXiv:2112.03663 using some concrete examples in the case of $D=105$.

math.NT↗

arXiv2022

Modeling Random Walks to Infinity on Primes in $\mathbb{Z}[\sqrt{2}]$

An interesting question, known as the Gaussian moat problem, asks whether it is possible to walk to infinity on Gaussian primes with steps of bounded length. Our work examines a similar situation in the real quadratic integer ring $\mathbb{Z}[\sqrt{2}]$ whose primes cluster near the asymptotes $y = \pm x/\sqrt{2}$ as compared to Gaussian primes, which cluster near the origin. We construct a probabilistic model of primes in $\mathbb{Z}[\sqrt{2}]$ by applying the prime number theorem and a combinatorial theorem for counting the number of lattice points whose absolute values of their norms are at most $r^2$. We then prove that it is impossible to walk to infinity if the walk remains within some bounded distance from the asymptotes. Lastly, we perform a few moat calculations to show that the longest walk is likely to stay close to the asymptotes; hence, we conjecture that there is no walk to infinity on $\mathbb{Z}[\sqrt{2}]$ primes with steps of bounded length.

math.NT↗

arXiv2022

Benfordness of the Generalized Gamma Distribution

The generalized gamma distribution shows up in many problems related to engineering, hydrology as well as survival analysis. Earlier work has been done that estimated the deviation of the exponential and the Weibull distribution from Benford's Law. We give a mathematical explanation for the Benfordness of the generalized gamma distribution and present a measure for the deviation of the generalized gamma distribution from the Benford distribution.

math.PR↗

arXiv2022

$k$-Diophantine $m$-tuples in Finite Fields

In this paper, we define a $k$-Diophantine $m$-tuple to be a set of $m$ positive integers such that the product of any $k$ distinct positive integers is one less than a perfect square. We study these sets in finite fields $\mathbb{F}_p$ for odd prime $p$ and guarantee the existence of a $k$-Diophantine m-tuple provided $p$ is larger than some explicit lower bound. We also give a formula for the number of 3-Diophantine triples in $\mathbb{F}_p$ as well as an asymptotic formula for the number of $k$-Diophantine $k$-tuples.

math.NT↗

arXiv2021

Connections of Class Numbers to the Group Structure of Generalized Pythagorean Triples

Two well-studied Diophantine equations are those of Pythagorean triples and elliptic curves; for the first, we have a parametrization through rational points on the unit circle, and for the second we have a structure theorem for the group of rational solutions. Recently Yekutieli discussed a connection between these two problems and described the group structure of Pythagorean triples and the number of triples for a given hypotenuse. We generalize these methods and results to Pell's equation. We find a similar group structure and count on the number of solutions for a given $z$ to $x^2 + Dy^2 = z^2$ when $D$ is 1 or 2 modulo 4 and the class group of $\mathbb{Q}[\sqrt{-D}]$ is a free $\mathbb{Z}_2$ module, which always happens if the class number is at most 2. We give examples of when the results hold for a class number greater than 2, as well as an example with different behavior when the class group does not have this structure.

math.NT↗

arXiv2021

Irreducibility over the Max-Min Semiring

For sets $A, B\subset \mathbb N$, their sumset is $A + B := \{a+b: a\in A, b\in B\}$. If we cannot write a set $C$ as $C = A+B$ with $|A|, |B|\geq 2$, then we say that $C$ is $\textit{irreducible}$. The question of whether a given set $C$ is irreducible arises naturally in additive combinatorics. Equivalently, we can formulate this question as one about the irreducibility of boolean polynomials, which has been discussed in previous work by K. H. Kim and F. W. Roush (2005) and Y. Shitov (2014). We prove results about the irreducibility of polynomials and power series over the max-min semiring, a natural generalization of the boolean polynomials. We use combinatorial and probabilistic methods to prove that almost all polynomials are irreducible over the max-min semiring, generalizing work of Y. Shitov (2014) and proving a 2011 conjecture by D. L. Applegate, M. Le Brun, and N. J. A. Sloane. Furthermore, we use measure-theoretic methods and apply Borel's result on normal numbers to prove that almost all power series are asymptotically irreducible over the max-min semiring. This result generalizes work of E. Wirsing (1953).

math.CO↗

arXiv2021

The Real Schwarz Lemma: The Sequel

A decade ago, when teaching complex analysis, the third named author posed the question on whether or not there is an analogue to the Schwarz lemma for real analytic functions. This led to the note [MT], indicating that it is possible to have a real analytic automorphism $f$ of $(-1,1)$ with $f'(0)$ arbitrarily large. In this note we provide other families with this property, and moreover show that we can always find such a function so that $f'(0)$ equals any desired real number. We end with some questions on related problems.

math.CV↗

arXiv2021

The Generalized Bergman Game

Every positive integer may be written uniquely as a base-$β$ decomposition--that is a legal sum of powers of $β$--where $β$ is the dominating root of a non-increasing positive linear recurrence sequence. Guided by earlier work on a two-player game which produces the Zeckendorf Decomposition of an integer (see [Bai+19]), we define a broad class of two-player games played on an infinite tuple of non-negative integers which decompose a positive integer into its base-$β$ expansion. We call this game the Generalized Bergman Game. We prove that the longest possible Generalized Bergman game on an initial state $S$ with $n$ summands terminates in $Θ(n^2)$ time, and we also prove that the shortest possible Generalized Bergman game on an initial state terminates between $Ω(n)$ and $O(n^2)$ time. We also show a linear bound on the maximum length of the tuple used throughout the game.

math.NT↗

arXiv2021

Limiting Spectral Distributions of Families of Block Matrix Ensembles

We introduce a new matrix operation on a pair of matrices, $\text{swirl}(A,X),$ and discuss its implications on the limiting spectral distribution. In a special case, the resultant ensemble converges almost surely to the Rayleigh distribution. In proving this, we provide a novel combinatorial proof that the random matrix ensemble of circulant Hankel matrices converges almost surely to the Rayleigh distribution, using the method of moments.

math.PR↗

arXiv2021

Completeness of Positive Linear Recurrence Sequences

A sequence of positive integers is complete if every positive integer is a sum of distinct terms. A positive linear recurrence sequence (PLRS) is a sequence defined by a homogeneous linear recurrence relation with nonnegative coefficients of the form $H_{n+1} = c_1 H_n + \cdots + c_L H_{n-L+1}$ and a particular set of initial conditions. We seek to classify various PLRS's by completeness. With results on how completeness is affected by modifying the recurrence coefficients of a PLRS, we completely characterize completeness of several families of PLRS's as well as conjecturing criteria for more general families. Our primary method is applying Brown's criterion, which says that an increasing sequence $\{H_n\}_{n = 1}^{\infty}$ is complete if and only if $H_1 = 1$ and $H_{n + 1} \leq 1 + \sum_{i = 1}^n H_i$. %A survey of these results can be found in \cite{BHLLMT}. Finally, we adopt previous analytic work on PLRS's to find a more efficient way to check completeness. Specifically, the characteristic polynomial of any PLRS has exactly one positive root; by bounding the size of this root, the majority of sequences may be classified as complete or incomplete. Additionally, we show there exists an indeterminate region where the principal root does not reveal any information on completeness. We have conjectured precise bounds for this region.

math.CO↗

arXiv2021

On Zeckendorf Related Partitions Using the Lucas Sequence

Zeckendorf proved that every positive integer has a unique partition as a sum of non-consecutive Fibonacci numbers. Similarly, every natural number can be partitioned into a sum of non-consecutive terms of the Lucas sequence, although such partitions need not be unique. In this paper, we prove that a natural number can have at most two distinct non-consecutive partitions in the Lucas sequence, find all positive integers with a fixed term in their partition, and calculate the limiting value of the proportion of natural numbers that are not uniquely partitioned into the sum of non-consecutive terms in the Lucas sequence.

math.NT↗

arXiv2021

Distinct Angle Problems and Variants

The Erdős distinct distance problem is a ubiquitous problem in discrete geometry. Less well known is Erdős' distinct angle problem, the problem of finding the minimum number of distinct angles between $n$ non-collinear points in the plane. The standard problem is already well understood. However, it admits many of the same variants as the distinct distance problem, many of which are unstudied. We provide upper and lower bounds on a broad class of distinct angle problems. We show that the number of distinct angles formed by $n$ points in general position is $O(n^{\log_2(7)})$, providing the first non-trivial bound for this quantity. We introduce a new class of asymptotically optimal point configurations with no four cocircular points. Then, we analyze the sensitivity of asymptotically optimal point sets to perturbation, yielding a much broader class of asymptotically optimal configurations. In higher dimensions we show that a variant of Lenz's construction admits fewer distinct angles than the optimal configurations in two dimensions. We also show that the minimum size of a maximal subset of $n$ points in general position admitting only unique angles is $Ω(n^{1/5})$ and $O(n^{\log_2(7)/3})$. We also provide bounds on the partite variants of the standard distinct angle problem.

math.CO↗

arXiv2021

Generalizing the Distribution of Missing Sums in Sumsets

Given a finite set of integers $A$, its sumset is $A+A:= \{a_i+a_j \mid a_i,a_j\in A\}$. We examine $|A+A|$ as a random variable, where $A\subset I_n = [0,n-1]$, the set of integers from 0 to $n-1$, so that each element of $I_n$ is in $A$ with a fixed probability $p \in (0,1)$. Recently, Martin and O'Bryant studied the case in which $p=1/2$ and found a closed form for $\mathbb{E}[|A+A|]$. Lazarev, Miller, and O'Bryant extended the result to find a numerical estimate for $\text{Var}(|A+A|)$ and bounds on the number of missing sums in $A+A$, $m_{n\,;\,p}(k) := \mathbb{P}(2n-1-|A+A|=k)$. Their primary tool was a graph-theoretic framework which we now generalize to provide a closed form for $\mathbb{E}[|A+A|]$ and $\text{Var}(|A+A|)$ for all $p\in (0,1)$ and establish good bounds for $\mathbb{E}[|A+A|]$ and $m_{n\,;\,p}(k)$. We continue to investigate $m_{n\,;\,p}(k)$ by studying $m_p(k) = \lim_{n\to\infty}m_{n\,;\,p}(k)$, proven to exist by Zhao. Lazarev, Miller, and O'Bryant proved that, for $p=1/2$, $m_{1/2}(6)>m_{1/2}(7) m_{p}(1)<m_{p}(2)$. Finally, we extend the graph-theoretic framework originally introduced by Lazarev, Miller, and O'Bryant to correlated sumsets $A+B$ where $B$ is correlated to $A$ by the probabilities $\mathbb{P}(i\in B \mid i\in A) = p_1$ and $\mathbb{P}(i\in B \mid i\not\in A) = p_2$. We provide some preliminary results using the extension of this framework.

math.NT↗

arXiv2021

Generalizing Zeckendorf's Theorem to Homogeneous Linear Recurrences, I

Zeckendorf's theorem states that every positive integer can be written uniquely as the sum of non-consecutive shifted Fibonacci numbers $\{F_n\}$, where we take $F_1=1$ and $F_2=2$. This has been generalized for any Positive Linear Recurrence Sequence (PLRS), which informally is a sequence satisfying a homogeneous linear recurrence with a positive leading coefficient and non-negative integer coefficients. These decompositions are generalizations of base $B$ decompositions. In this and the follow-up paper, we provide two approaches to investigate linear recurrences with leading coefficient zero, followed by non-negative integer coefficients, with differences between indices relatively prime (abbreviated ZLRR). The first approach involves generalizing the definition of a legal decomposition for a PLRS found in Koloğlu, Kopp, Miller and Wang. We prove that every positive integer $N$ has a legal decomposition for any ZLRR using the greedy algorithm. We also show that a specific family of ZLRRs loses uniqueness of decompositions. The second approach converts a ZLRR to a PLRR that has the same growth rate. We develop the Zeroing Algorithm, a powerful helper tool for analyzing the behavior of linear recurrence sequences. We use it to prove a very general result that guarantees the possibility of conversion between certain recurrences, and develop a method to quickly determine whether certain sequences diverge to $+\infty$ or $-\infty$, given any real initial values. This paper investigates the first approach.

math.NT↗

arXiv2021

Generalizing Zeckendorf's Theorem to Homogeneous Linear Recurrences, II

Zeckendorf's theorem states that every positive integer can be written uniquely as the sum of non-consecutive shifted Fibonacci numbers $\{F_n\}$, where we take $F_1=1$ and $F_2=2$. This has been generalized for any Positive Linear Recurrence Sequence (PLRS), which informally is a sequence satisfying a homogeneous linear recurrence with a positive leading coefficient and non-negative integer coefficients. In this and the preceding paper we provide two approaches to investigate linear recurrences with leading coefficient zero, followed by non-negative integer coefficients, with differences between indices relatively prime (abbreviated ZLRR), via two different approaches. The first approach involves generalizing the definition of a legal decomposition for a PLRS found in Koloğlu, Kopp, Miller and Wang. We prove that every positive integer $N$ has a legal decomposition for any ZLRR using the greedy algorithm. We also show that a specific family of ZLRRs lost uniqueness of decompositions. The second approach converts a ZLRR to a PLRR that has the same growth rate. We develop the Zeroing Algorithm, a powerful helper tool for analyzing the behavior of linear recurrence sequences. We use it to prove a very general result that guarantees the possibility of conversion between certain recurrences, and develop a method to quickly determine whether a sequence diverges to $+\infty$ or $-\infty$, given any real initial values. This paper investigates the second approach.

math.NT↗

arXiv2021

When Rooks Miss: Probability through Chess

A famous (and hard) chess problem asks what is the maximum number of safe squares possible in placing $n$ queens on an $n\times n$ board. We examine related problems from placing $n$ rooks. We prove that as $n\to\infty$, the probability rapidly tends to 1 that the fraction of safe squares from a random placement converges to $1/e^2$. Our interest in the problem is showing how to view the involved algebra to obtain the simple, closed form limiting fraction. In particular, we see the power of many of the key concepts in probability: binary indicator variables, linearity of expectation, variances and covariances, Chebyshev's inequality, and Stirling's formula.

math.PR↗

arXiv2021

Biases in Moments of the Dirichlet Coefficients in One- and Two-Parameter Families of Elliptic Curves

We study one-parameter families of elliptic curves over $\mathbb{Q}(T)$, which are of the form $y^2=x^3+A(T)x+B(T)$, with non-constant $j$-invariant. We define the $r$\textsuperscript{th} moment of an elliptic curve to be $A_{r,E}(p) := \frac1{p} \sum_{t \bmod p} a_t(p)^r$, where $a_t(p)$ is $p$ minus the number of solutions to $y^2 = x^3 + A(t)x + B(t) \bmod p$. Rosen and Silverman showed biases in the first moment equal the rank of the Mordell-Weil group of rational solutions. Michel proved that $pA_{2,E}(p)=p^2+O(p^{3/2})$. Based on several special families where computations can be done in closed form, Miller in his thesis conjectured that the largest lower-order term in the second moment that does not average to $0$ is on average negative. He further showed that such a negative bias has implications in the distribution of zeros of the elliptic curve $L$-function near the central point. To date, evidence for this conjecture is limited to special families. In this paper, we explore the first and second moments of some one- and two-parameter families of elliptic curves, looking to see if the biases persist and exploring the consequence these have on fundamental properties of elliptic curves. We observe that in all of the one- and two-parameter families where we can compute in closed form that the first term that does not average to zero in the second-moment expansion of the Dirichlet coefficients has a negative average. In addition to studying some additional families where the calculations can be done in closed form, we also systematically investigate families of various rank. These are the first general tests of the conjecture; while we cannot in general obtain closed form solutions, we discuss computations which support or contradict the conjecture. We then generalize to higher moments, and see evidence that the bias continues in the even moments.

math.NT↗