arXiv subjects

Steven J. Miller

Publications and source records attributed to Steven J. Miller.

At least 19 recordsLinked to original sources

arXiv2026

Generalizing a Pair of Diophantine Equations

For coprime integers $a$ and $b$, it is known that exactly one of the two Diophantine equations $$ ax+by\ =\ \frac{(a-1)(b-1)}{2} \qquad\text{and}\qquad 1+ax+by\ =\ \frac{(a-1)(b-1)}{2} $$ admits a nonnegative integer solution, and that this solution is unique. We first generalize this result by replacing the right-hand side with an arbitrary integer $m$ and its complement $ab-a-b-m$. This framework enables us to study the existence and uniqueness of nonnegative integer solutions to $$ ax+by\ =\ \frac{(a-1)(b-1)}{k} \qquad\text{and}\qquad 1+ax+by\ =\ \frac{(a-1)(b-1)}{k}, $$ where $k$ is a fixed positive integer. We then obtain explicit results when $a$ and $b$ are consecutive Fibonacci numbers. Finally, we examine the original pair of equations in several particular settings, including when $b\equiv \pm1\mod a$, when $b$ is replaced by a higher power, and when the parameters are squared.

math.NT↗

arXiv2026

Consequences of Matrices Sharing Eigenvalues and Eigenvectors

While studying for a linear algebra final, the first named author prepared some test questions for herself to see how well she understood the material, and asked the second named author: \emph{If $A$ and $A^T$ have the same eigenvalues and eigenvectors, is $A$ a symmetric matrix?} We show how this excellent question is a great springboard to related questions, in particular when do equal eigenvalues and eigenvectors imply the matrices are, if not equal, at least closely related (such as similar or the transpose/complex conjugate transpose of each other)? The answer depends on how we interpret the question, and provides a great opportunity to talk about creating good questions. In particular, we characterize matrices $A$ for which the transpose or conjugate transpose shares the same eigenvectors (regardless of eigenvalues) and, for each eigenvalue, the same eigenpair (equivalently, the same eigenspace). Thus, a square matrix $A$ is Hermitian if and only if $A^*$ has the same eigenpairs as $A$; moreover, if $A$ is a real matrix with real eigenvalues and $A^T$ has the same eigenvectors as $A$, then $A$ is symmetric.

math.GM↗

arXiv2026

Braess' Paradox in Uniform Affine Grid Networks

Braess' Paradox is the phenomenon in which adding an edge to a congestion network increases total travel time. We study the paradox in directed rectangular grids where every edge shares the latency function $\ell(x)=ax+b$ and an added chord has latency $\ell_{*}(x) = cx+d$, where $a > 0$ and $b,c,d \ge 0$. Using an analogy with electrical networks, we bound the change in total travel time. We then give a necessary and sufficient condition for a chord to induce the paradox for some choice of nonnegative coefficients and compute the exact proportion of such chords in all grids with dimensions at most $100$. Such chords are scarce, and the fraction is maximized near an aspect ratio of $2:1$. We then improve the established $4/3$ upper bound on the Braess Ratio to one depending only on the grid dimensions, approaching $1.207$ on squares and $4/3$ on thin grids. Finally, we prove any ratio-maximizing chord must have zero latency.

math.CO↗

arXiv2026

Counting Schreier Sets Under Neighborhood Conditions

We count Schreier sets that satisfy a neighborhood condition, including $k$-clustered, $k$-consecutive-free, $k$-neighbored, $k$-isolated, and closed under integral $2$-averages. For the first four conditions, we determine the initial counts and prove linear recurrence relations. For the last condition, we prove a recurrence that involves the divisor counting function.

math.CO↗

arXiv2026

Bounding The Number of Zeros Near the Central Point in Families of Cuspidal Newforms

We study low-lying zeros in families of even holomorphic cuspidal newforms of fixed weight and prime level, with particular emphasis on the number of forms having a zero in a prescribed normalized window about the central point and on the distribution of the number of such zeros among the forms. We quantify the number of forms having at least one zero in the window and study the distribution of the number of zeros in that window among the forms. Assuming the Generalized Riemann Hypothesis, we obtain new lower bounds for the number of forms having a low-lying zero. We first prove that, along an infinite sequence of prime levels $N$, the number of such forms is $\gg N^{7/8}\log N$. We then use higher centered moments and an appropriate test function to show that a positive proportion of the family has a zero in a prescribed window. Finally, we obtain polynomial upper-tail bounds for the number of zeros occurring there and show that a positive proportion of forms have a bounded, nonzero number of low-lying zeros.

math.NT↗

arXiv2026

On the Existence of Hyperelliptic Curves over $\mathbb{Q}(T)$ with Certain Jacobian Ranks

Let $y^2 = f(x,T)$ be a hyperelliptic curve of genus $g\geq 1$, defined over $\mathbb{Q}(T)$. We prove the existence of infinitely many imaginary hyperelliptic curves with a fixed genus $g$ having a certain rank for $5\leq r\leq 4g+2$, and a similar result for real hyperelliptic curves with a fixed genus $g$ having a certain rank for $6\leq r\leq 4g+4$. We begin by constructing such curves and prove the rank using two methods. First, we apply the generalized Nagao's conjecture, which relates the first moment and the rank of the Jacobian variety $J_\mathcal{X}(\mathbb{Q}(T))$, and that the conjecture holds for our curves, making the result unconditional. Furthermore, we explicitly construct rational points in the Mordell-Weil group and use Shioda-Tate to prove that the rank is equal to $r$.

math.NT↗

arXiv2026

The Fibonacci Rectangle Game: Two First-Move Classes and a Triangle-Induced Choice

A square-adjoining rectangle game generates the Fibonacci numbers and the Fibonacci spiral from a simple geometric rule. If one starts from a square, the four possible first moves are all equivalent by rotation. If one starts instead from a non-square rectangle, there are still four geometric placements for the first square, but they split into exactly two equivalence classes: long-side-first and short-side-first. We show that both classes are governed by the same Fibonacci-type recursion with different initial conditions, and that in both cases the successive aspect ratios converge to the golden ratio (phi). We then add a brief geometric remark: the Hypotenuse-Axis Intercept (HAI) construction from a right triangle produces a natural ordered seed whose outward and inward branches determine precisely those two first-move classes.

math.CO↗

arXiv2026

On Zeckendorf-Niven numbers and arithmetic progressions

A positive integer is Zeckendorf-Niven (respectively, Lucas-Niven) if it is divisible by the number of summands in its Zeckendorf decomposition (respectively, Lucas decomposition). We show that there exist infinitely many Zeckendorf-Niven numbers and Lucas-Niven numbers in every arithmetic progression. Furthermore, we provide bounds on the maximum number of consecutive Zeckendorf-Niven terms in certain arithmetic progressions.

math.NT↗

arXiv2026

Spectral Properties of Dense Barab\'asi-Albert Graphs

Preferential attachment graphs model networks whose growth produces highly uneven degree distributions, describing many real-world systems. Their adjacency spectra are important because they allow graph-theoretic questions to be studied through the eigenvalues of matrices. We analyze the adjacency matrix of a dense Barab\'asi-Albert (B-A) multigraph, where the number of edges added at each step is proportional to the final number of vertices. First, we compute the large-$n$ limit of the expected adjacency matrix and show that it is described by a rank-one limiting kernel, viewed as a continuous analogue of the adjacency matrix. After centering and scaling, the fluctuations form a random matrix with a computable variance profile. Using the quadratic vector equation approach, we derive the limiting bulk spectral distribution. We also determine the asymptotic location of the leading eigenvalue generated by the rank-one mean component.

math.PR↗

arXiv2026

Fibonacci Numbers and Vieta Jumping for a Rational Diophantine Equation

We study the Diophantine equation $\displaystyle{\tfrac{a+1}{b} + \tfrac{b+1}{a} \ = \ k}$, where $k$ is an integer. Using Vieta jumping, we completely classify all positive integer pairs $(a, \, b)$. We prove that the associated integer value $k$ can only be $3$ or $4$. The corresponding solution pairs $(a,\,b)$ are related to the classical Fibonacci numbers. As a consequence, the quantity $\frac{a+b}{\gcd(a, \,b)^2}$ takes only the values $1, \, 2, \, 3$ and $5$. This reveals an unexpected connection between a simple rational Diophantine condition, Vieta jumping, and Fibonacci numbers.

math.NT↗

arXiv2025

Problems Regarding a Pair of Diophantine Equations

For two relatively prime positive integers $a, b\in \mathbb{N}$, it is known that exactly one of the two Diophantine equations $$ax + by \ =\ \frac{(a-1)(b-1)}{2}\ \mbox{ and }\ 1 + ax + by \ =\ \frac{(a-1)(b-1)}{2}$$ has a nonnegative integral solution $(x, y)$. Furthermore, the solution is unique. In this note, we summarize recent results and some new ones on the solution of the two equations and provide an overview of problems for future investigation, some of which were presented at the 2025 International Conference on Class Groups of Number Fields and Related Topics.

math.NT↗

arXiv2025

On a Roll Again: Analysis of a Dice Removal Game

Suppose we have $n$ dice, each with $s$ faces (assume $s\geq n$). On the first turn, roll all of them, and remove from play those that rolled an $n$. Roll all of the remaining dice. In general, if at a certain turn you are left with $k$ dice, roll all of them and remove from play those that rolled a $k$. The game ends when you are left with no dice to roll. For $n,s \in \mathbb{N} \setminus \{0\}$ such that $s \geq n$, let $Y_n^s$ be the random variable for the number of turns to finish the game rolling $n$ dice with $s$ faces. We find recursive and non-recursive solutions for $\mathbb{E}(Y_n^{s})$ and $\mathrm{Var}(Y_n^{s})$, and bounds for both values. Moreover, we show that $Y_n^{s}$ can also be modeled as the maximum of a sequence of i.i.d. geometrically distributed random variables. Although, as far as we know, this game hasn't been studied before, similar problems have.

math.PR↗

arXiv2025

Egg Drop Problems: They Are All They Are Cracked Up To Be!

We illustrate how to invite and excite students about research by exploring higher-dimensional generalizations of the classical egg drop problem, in which the goal is to locate a critical breaking point using the fewest number of trials. Beginning with the one-dimensional case, we prove that with $k$ eggs and $N$ floors, the minimal number of drops in the worst case satisfies $P_1(k) \leq \lceil k N^{1/k} \rceil$. We then extend the recursive algorithm to two and three dimensions, proving similar formulas: $P_2(k) \leq \lceil (k-1)(M+N)^{1/(k-1)} \rceil $ in 2D and $P_3(k) \leq \lceil (k-2)(L+M+N)^{1/(k-2)} \rceil$ in 3D, and conjecture a general formula for the $d$-dimensional case. Beyond the critical point problems, we then study the critical line problems, where the breaking condition occurs along $x+y=V$ (with slope $-1$) or, more generally, $\alpha x+\beta y=V$ (with the slope of the line unknown). We discuss how one frequently has to pivot from the original problem, which is intractable, to something that can be solved; in our case, using induction and recursion, two standard proof techniques.

math.HO↗

arXiv2025

Signal recovery using Gabor frames

We present a novel probabilistic framework for the recovery of discrete signals with missing data, extending classical Fourier-based methods. While prior results, such as those of Donoho and Stark; see also Logan's method, guarantee exact recovery under strict deterministic sparsity constraints, they do not account for stochastic patterns of data loss. Our approach combines a row-wise Gabor transform with a probabilistic model for missing frequencies, establishing near-certain recovery when losses occur randomly. The key innovation is a maximal row-support criterion that allows unique reconstruction with high probability, even when the overall signal support significantly exceeds classical bounds. Specifically, we show that if missing frequencies are independently distributed according to a binomial law, the probability of exact recovery converges to $1$ as the signal size grows. This provides, to our knowledge, the first rigorous probabilistic recovery guarantee exploiting row-wise signal structure. Our framework offers new insights into the interplay between sparsity, transform structure, and stochastic loss, with immediate implications for communications, imaging, and data compression. It also opens avenues for future research, including extensions to higher-dimensional signals, adaptive transforms, and more general probabilistic loss models, potentially enabling even more robust recovery guarantees.

math.CA↗

arXiv2025

General Recurrence Multidimensional Zeckendorf Representations

We present a multidimensional generalization of Zeckendorf's Theorem (any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers) to a large family of linear recurrences. This extends work of Anderson and Bicknell-Johnson in the multi-dimensional case when the underlying recurrence is the same as the Fibonacci one. Our extension applies to linear recurrence relations defined by vectors $\vec{\mathbf{c}} = (c_1, c_2, \ldots, c_k)$ such that $c_1\geq c_2\geq\cdots \geq c_k$ and where $c_k = 1$. Under these conditions, we prove that every integer vector in $\mathbb{Z}^{k-1}$ admits a unique $\vec{\mathbf{c}}$-satisfying representation ($\vec{\mathbf{c}}$-SR) as a linear combination of vectors, $(\vec{\mathbf{X}}_n)_{n\in \mathbb{Z}}$ defined for every $n\in \mathbb{Z}$ by initially by zero and standard unit vectors and then the recursion $$\vec{\mathbf{X}}_{n} := c_1\vec{\mathbf{X}}_{n -1} + c_2\vec{\mathbf{X}}_{n - 2} + \cdots + c_k\vec{\mathbf{X}}_{n-k}.$$ To establish this, we introduce carrying and borrowing operations that use the defining recursion to transform any $\vec{\mathbf{c}}$ representation into a $\vec{\mathbf{c}}$-SR while preserving the underlying vector. Then, by establishing bijections with properties of scalar Positive Linear Recurrence Sequences (PLRS), we prove that these multidimensional decompositions inherit various properties, such as the number of summands exhibits Gaussian behavior and summand minimality of $\vec{\mathbf{c}}$-SRs over all all $\vec{\mathbf{c}}$-representations.

math.NT↗

arXiv2025

Properties of Multidimensional Vector Zeckendorf Representations

Zeckendorf's Theorem says that for all $k \geq 3$, every nonnegative integer has a unique $k$-Zeckendorf representation as a sum of distinct $k$-bonacci numbers, where no $k$ consecutive $k$-bonacci numbers are present in the representation. Anderson and Bicknell-Johnson extend this result to the multidimensional context: letting the $k$-bonacci vectors $\vec{\mathbf{X}}_i \in \mathbb{Z}^{k-1}$ be given by $\vec{\mathbf{X}}_0=\vec{\mathbf{0}}$, $\vec{\mathbf{X}}_{-i}=\vec{\mathbf{e}}_i$ for $1 \leq i \leq k-1$, and $\vec{\mathbf{X}}_n=\sum_{i=1}^k \vec{\mathbf{X}}_{n-i}$ for all $n \in \mathbb{Z}$, they show that for all $k \geq 3$, every $\vec{\mathbf{v}} \in \mathbb{Z}^{k-1}$ has a unique $k$-bonacci vector Zeckendorf representation, a sum of distinct $k$-bonacci vectors where no $k$ consecutive $k$-bonacci vectors are present in the representation. Their proof provides an inductive algorithm for finding such representations. We present two improved algorithms for finding the $k$-bonacci vector Zeckendorf representation of $\vec{\mathbf{v}}$ and analyze their relative efficiency. We utilize a projection map $S_n:\mathbb Z^{k-1} \to \mathbb Z_{\geq 0}$, introduced in Anderson and Bicknell-Johnson work, that reduces the study of $k$-bonacci vector representations to the setting of $k$-bonacci number representations, provided a lower bound is established for the most negatively indexed $k$-bonacci vector present in the $k$-bonacci vector Zeckendorf representation of $\vec{\mathbf{v}}$. Using this map and a bijection between $\mathbb Z^{k-1}$ and $\mathbb Z_{\geq 0}$, we further show that the number of and gaps between summands in $k$-bonacci vector Zeckendorf representations exhibit the same properties as those in $k$-Zeckendorf representations and that $k$-bonacci vector Zeckendorf representations exhibit summand minimality.

math.NT↗

arXiv2025

Additional Constructions of Sequences of Alternating Sum and Difference Dominated Sets

A More Sums Than Differences (MSTD) set is a finite set of integers $A$ where the cardinality of its sumset, $A+A$, is greater than the cardinality of its difference set, $A-A$. We address a problem posed by Samuel Allen Alexander that asks whether there exists an infinite sequence of sets alternating between being MSTD and More Differences Than Sums (MDTS), where each set properly contains the previous. While a companion paper resolved this using `filling in' techniques, we solve the more challenging `non-filling-in' version, where any missing integer between a set's minimum and maximum elements remains missing in all subsequent sets.

math.NT↗

arXiv2025

Centered Moments of Weighted One-Level Densities of $GL(2)$ $L$-Functions

Katz and Sarnak conjectured that the behavior of zeros near the central point of any family of $L$-functions is well-modeled by the behavior of eigenvalues near $1$ of some classical compact group (either the symplectic, unitary, or even, odd, or full orthogonal group). In 2018, Knightly and Reno proved that the symmetry group can vary depending on how the $L$-functions in the family are weighted. They observed both orthogonal and symplectic symmetry in the one-level densities of families of cuspidal newform $L$-functions for different choices of weights. We observe the same dependence of symmetry on weights in the $n^{\text{th}}$ centered moments of these one-level densities, for smooth test functions whose Fourier transforms are supported in $\left(-\frac{1}{2n}, \frac{1}{2n}\right)$. To treat the new terms that emerge in our $n$-level calculations when $n>1$, i.e., the cross terms that emerge from $n$-fold products of primes rather than individual primes, we generalize Knightly and Reno's weighted trace formula from primes to arbitrary positive integers. We then perform a delicate analysis of these cross terms to distinguish their contributions to the main and error terms of the $n^{\text{th}}$ centered moments. The final novelty here is an elementary combinatorial trick that we use to rewrite the main number theoretic terms arising from our analysis, facilitating comparisons with random matrix theory.

math.NT↗