arXiv subjects

Steven J. Miller

Publications and source records attributed to Steven J. Miller.

At least 37 records · Page 2Linked to original sources

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↗

arXiv2026

Comparing Left and Right Quotient Sets in Groups

For a finite subset $A$ of a group $G$, we define the right quotient set and the left quotient set of $A$, respectively, as $AA^{-1} := \{a_1a_2^{-1}:a_1,a_2\in A\}$, $A^{-1}A := \{a_1^{-1}a_2:a_1,a_2\in A\}$. While the right and left quotient sets are equal if $G$ is abelian, subtleties arise when $G$ is a nonabelian group, where the cardinality difference $|AA^{-1}| - |A^{-1}A|$ may be take on arbitrarily large values. Using the results of Martin and O'Bryant on the cardinality differences of sum sets and difference sets in $\mathbb{Z}$, we prove in the infinite dihedral group, $D_\infty \cong \mathbb{Z} \rtimes \mathbb{Z}/2\mathbb{Z}$, every integer difference is achievable. Further, we prove that in $F_2$, the free group on $2$ generators, an integer difference is achievable if and only if that integer is even, and we explicitly construct subsets of $F_2$ that achieve every even integer. We further determine the minimum cardinality of $A \subset G$ so that the difference between the cardinalities of the left and right quotient sets is nonzero, depending on the existence of order $2$ elements in $G$. To prove these results, we construct difference graphs $D_A$ and $D_{A^{-1}}$ which encode equality, respectively, in the right and left quotient sets. We observe a bijection from edges in $D_A$ to edges in $D_{A^{-1}}$ and count connected components in order to obtain our results on cardinality differences $|AA^{-1}| - |A^{-1}A|$.

math.NT↗

arXiv2026

The Ordered Zeckendorf Game

We introduce and analyze the ordered Zeckendorf game, a novel combinatorial two-player game inspired by Zeckendorf's Theorem, which guarantees a unique decomposition of every positive integer as a sum of non-consecutive Fibonacci numbers. Building on the original Zeckendorf game\ -- previously studied in the context of unordered multisets\ -- we impose a new constraint: all moves must respect the order of summands. The result is a richer and more nuanced strategic landscape that significantly alters game dynamics. Unlike the classical version, where Player 2 has a dominant strategy for all $n > 2$, our ordered variant reveals a more balanced and unpredictable structure. In particular, we find that Player 1 wins for nearly all values $n \leq 25$, with a single exception at $n = 18$. This shift in strategic outcomes is driven by our game's key features: adjacency constraints that limit allowable merges and splits to neighboring terms, and the introduction of a switching move that reorders pairs. We prove that the game always terminates in the Zeckendorf decomposition\ -- now in ascending order\ -- by constructing a strictly decreasing monovariant. We further establish bounds on game complexity: the shortest possible game has length exactly $n - Z(n)$, where $Z(n)$ is the number of summands in the Zeckendorf decomposition of $n$, while the longest game exhibits quadratic growth, with $M(n) \sim \frac{n^2}{2}$ as $n \to \infty$. Empirical simulations suggest that random game trajectories exhibit log-normal convergence in their move distributions. Overall, the ordered Zeckendorf game enriches the landscape of number-theoretic games, posing new algorithmic challenges and offering fertile ground for future exploration into strategic complexity, probabilistic behavior, and generalizations to other recurrence relations.

math.NT↗

arXiv2026

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↗

arXiv2026

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↗

arXiv2026

On a Pair of Diophantine Equations

For relatively prime natural numbers $a$ and $b$, we study the two equations $ax+by = (a-1)(b-1)/2$ and $ax+by+1= (a-1)(b-1)/2$, which arise from the study of cyclotomic polynomials. Previous work showed that exactly one equation has a nonnegative solution, and the solution is unique. Our first result gives criteria to determine which equation is used for a given pair $(a,b)$. We then use the criteria to study the sequence of equations used by the pair $(a_n/\gcd{(a_n, a_{n+1})}, a_{n+1}/\gcd{(a_n, a_{n+1})})$ from several special sequences $(a_n)_{n\geq 1}$. Finally, fixing $k \in \mathbb{N}$, we investigate the periodicity of the sequence of equations used by the pair $(k/\gcd{(k, n)}, n/\gcd{(k, n)})$ as $n$ increases.

math.NT↗

arXiv2026

Applications of Improvements to the Pythagorean Won-Loss Expectation in Optimizing Rosters

Bill James' Pythagorean formula has for decades done an excellent job estimating a baseball team's winning percentage from very little data: if the average runs scored and allowed are denoted respectively by ${\rm RS}$ and ${\rm RA}$, there is some $γ\approx 2$ such that the winning percentage is approximately ${\rm RS}^γ/ ({\rm RS}^γ+ {\rm RA}^γ)$. One use case is to determine the value of potential signings to the team, as it allows us to estimate how many more wins one obtains over a season given an estimated change in run production and concession. We summarize earlier work on the subject, and extend the earlier theoretical model of Miller (who assumed the home and away teams' runs arise from independent Weibull distributions with the same shape parameter $γ$; this has been observed to describe the observed run data well and yields a win probability equivalent to that of James' formula). We extend this work to model runs scored and allowed as being drawn from independent Weibull distributions with different shape parameters, and then consider the first and second moments to solve a system of four equations in the four unknowns. Doing so fits the training data better, yielding a higher winning percentage over the last 30 MLB seasons (1994 to 2023). This comes at a small cost as we no longer have a closed form expression for the win probability, but must evaluate a two-dimensional integral of Weibull distributions and numerically estimate the solutions to the system of equations. These are trivial to do with simple computational programs.

stat.AP↗

arXiv2026

The German Tank Problem with Multiple Factories

During the Second World War, estimates of the number of tanks deployed by Germany were critically needed. The Allies adopted a successful statistical approach to estimate this information: assume that the tanks are sequentially numbered starting from, say, 1, and ending at an unknown positive integer $N$. If we observe the numbers of $k$ tanks, then the best linear unbiased estimator for $N$ is $M(1+1/k)-1$ where $M$ is the maximum observed serial number. While this approach was successful, there are many more adversarial situations where the approach for the original German Tank Problem falls short. Typically the number of ``factories'' is a possibly unknown $l>1$, and tanks produced by different factories may have serial numbers in disjoint ranges that are often separated by unknown amounts. Clark, Gonye and Miller (CGM) presented an unbiased estimator for $N$ when the minimum serial number is unknown. So if one can identify which samples correspond to which factory, one can then estimate each factory's range using CGM's method, and sum them for an estimate of the rival's total productivity. We present a procedure to estimate the total productivity and prove that it is effective when $\log l/\log k$ is sufficiently small. In the final section, we show that if we have a small number of samples, we can make an estimator that performs orders of magnitude better when given additional information about the size of the gaps.

math.ST↗

arXiv2026

Phase Transitions for Sparse Random Sets Under Linear Forms

Let $A \subseteq \{0,1,\dots,N\}$ be a random set in which each element is included independently with probability $p=p(N)$. Fix an integer $h \geq 2$ and a linear form $$L(x_1,\dots,x_h) := u_1x_1 + \cdots + u_hx_h.$$ We study the random image set \begin{align*} L(A) = \left\{ L(a_1,\dots,a_h) : a_i \in A \right\}, \end{align*} inside the feasible interval of values of $L$ on $\{0,1,\dots,N\}^h$, as well as the associated representation counts. Our results exhibit two distinct threshold scales. First, there is a \emph{global} transition at $p(N) \asymp N^{-(h-1)/h}$ governing the size of $L(A)$: below this scale collisions are rare and $L(A)$ is sparse, while above it $L(A)$ contains nearly all feasible values. We give sharp asymptotics for the size of $L(A)$ in all regimes, including inside the critical window. Second, there is a \emph{local} transition at $p(N)\asymp N^{-(h-2)/(h-1)}$ governing multiplicities: for typical values in the bulk, the number of essentially distinct representations is asymptotically Poisson below this scale, and Poisson behavior fails above it. For $h \geq 3$ these scales are separated, yielding a regime in which $L(A)$ is already globally close to full while local multiplicities remain approximately Poisson. Our framework subsumes the classical sumset and difference-set models, as well as generalized sumsets of the form $sA-dA$, as special cases. Notably, after correcting its formulation, our global theorem settles the 2009 threshold conjecture of Hegarty-Miller on the behavior of these random images.

math.CO↗

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

Hyper-bishops, Hyper-rooks, and Hyper-queens: Percentage of Safe Squares on Higher Dimensional Chess Boards

The $n$ queens problem considers the maximum number of safe squares on an $n \times n$ chess board when placing $n$ queens; the answer is only known for small $n$. Miller, Sheng and Turek considered instead $n$ randomly placed rooks, proving the proportion of safe squares converges to $1/e^2$. We generalize and solve when randomly placing $n$ hyper-rooks and $n^{k-1}$ line-rooks on a $k$-dimensional board, using combinatorial and probabilistic methods, with the proportion of safe squares converging to $1/e^k$. We prove that the proportion of safe squares on an $n \times n$ board with bishops in 2 dimensions converges to $2/e^2$. This problem is significantly more interesting and difficult; while a rook attacks the same number of squares wherever it's placed, this is not so for bishops. We expand to the $k$-dimensional chessboard, defining line-bishops to attack along $2$-dimensional diagonals and hyper-bishops to attack in the $k-1$ dimensional subspace defined by its diagonals in the $k-2$ dimensional subspace. We then combine the movement of rooks and bishops to consider the movement of queens in 2 dimensions, as well as line-queens and hyper-queens in $k$ dimensions.

math.CO↗

arXiv2025

Variants of Conway Checkers and k-nacci Jumping

Conway Checkers is a game played with a checker placed in each square of the lower half of an infinite checkerboard. Pieces move by jumping over an adjacent checker, removing the checker jumped over. Conway showed that it is not possible to reach row 5 in finitely many moves by weighting each cell in the board by powers of the golden ratio such that no move increases the total weight. Other authors have considered the game played on many different boards, including generalising the standard game to higher dimensions. We work on a board of arbitrary dimension, where we allow a cell to hold multiple checkers and begin with m checkers on each cell. We derive an upper bound and a constructive lower bound on the height that can be reached, such that the upper bound almost never fails to be equal to the lower bound. We also consider the more general case where instead of jumping over 1 checker, each checker moves by jumping over k checkers, and again show the maximum height reachable lies within bounds that are almost always equal.

math.CO↗

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, $αx+βy=V$ (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

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

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

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

A Pair of Diophantine Equations and Fibonacci-Like Sequences

Given two relatively prime numbers $a$ and $b$, it is known that exactly one of the two Diophantine equations has a nonnegative integral solution $(x,y)$: $$ ax + by \ =\ \frac{(a-1)(b-1)}{2}\quad \mbox{ and }\quad 1 + ax + by \ =\ \frac{(a-1)(b-1)}{2}. $$ Furthermore, the solution is unique. This paper surveys recent results on finding the solution and determining which equation is used when $a$ and $b$ are taken from certain sequences. We contribute to the literature by finding $(x,y)$ when $a$ and $b$ are consecutive terms of sequences having the Fibonacci recurrence and arbitrary initial terms.

math.NT↗