arXiv subjects

Steven J. Miller

Publications and source records attributed to Steven J. Miller.

At least 19 recordsLinked to original sources

arXiv2026

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 $n\in \mathbb{Z}$ 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 $\vec{\mathbf{c}}$-representations.

math.NT↗

arXiv2026

Classification of crescent configurations

Let $n$ points be in crescent configuration in $\mathbb{R}^d$ if they lie in general position in $\mathbb{R}^d$ and determine $n-1$ distinct distances, such that for every $1 \leq i \leq n-1$ there is a distance that occurs exactly $i$ times. For the planar problem, Erdős conjectured that there exists $N$ such that no crescent configuration exists on $n$ points whenever $n\geq N$. In the same paper, he also gave constructions for $n$ up to 8, with $n=5$ due to Pomerance and $n=7,8$ due to Palásti, but nothing is yet known for $n \geq 9$. Most recently, Burt et al. proved that a crescent configuration on $n$ points exists in $\mathbb{R}^{n-2}$ for $n \geq 3$. In this paper, we classify planar crescent configurations by their complete distance-equality patterns, up to relabeling the vertices. We restrict attention to the strict distance classes defined below. There are three such classes on four vertices, all of which admit planar crescent realizations. On five vertices, there are $49$ strict classes, exactly $31$ of which admit planar crescent realizations. On six vertices, there are $6837$ strict classes, exactly six of which admit planar crescent realizations. The classification combines finite enumeration with exact algebraic verification of realizations and nonexistence. The distance-geometric criteria also apply in higher dimensions.

math.CO↗

arXiv2026

Experimental Mathematics: Discovering The Half-Derivative from First Principles

This expository paper grew out of two lectures in Miller's Math 409: The little Questions (Williams College, Fall 2026). We explore fractional derivatives by asking what a ``half-derivative'' might mean if applying it twice is to recover an ordinary derivative. Linearity and a natural guess for the half-derivative of a power function lead to a functional equation related to the classical factorial recurrence. Choosing the familiar Gamma-function extension gives the classical formula \[ \mathscr{D}^{1/2}(x^n) \ =\ \frac{Γ(n+1)}{Γ(n+\tfrac12)}x^{n-1/2}. \] We derive $Γ(n+1)=n!$, evaluate $Γ(1/2)=\sqrtπ$ using the Gaussian integral, and verify the half-derivative formula with an elementary integral. Along the way, we explore how the Gamma function generalizes factorials and binomial coefficients beyond their classical domains, encounter the double factorial, see why the usual product rule fails, and examine the half-derivative of $e^x$. We also explain why the original requirements do not determine a unique operator. The treatment emphasizes motivation, the benefits of experimentation through accessible calculations in test cases, rather than the general theory of fractional calculus.

math.HO↗

arXiv2026

Breaking Universality in the Lower Order Terms in the 1-level and 2-level Density of Holomorphic Cusp Newforms

The Katz--Sarnak density conjecture predicts that the normalized low-lying zeros of a family of $L$-functions ordered by analytic conductor $R$ converge, as $R \to \infty$, to the eigenvalue statistics near $1$ of a classical compact group, and that the limit depends on the family only through its symmetry type. We study the first correction to this limit for families of holomorphic cusp newforms of fixed weight $k$ and varying level $N$. Sharpening work of Miller on lower order terms in the one-level density, we evaluate the harmonically weighted one- and two-level densities with errors $O\left(\log^{-5}(R)\right)$ and $O\left(\log^{-2}(R)\right)$, and show that each of them already differs at order $\log^{-1}(R)$ from an explicit universal profile built out of $k$, the test functions, and $R$ alone. We compute that difference exactly. It is the product of a functional of the test functions with a single arithmetic invariant $κ_1(F)$ of the family, an explicit sum over the bounded prime factors of the level. Two families of newforms of the same weight and the same symmetry type, with conductors of the same size, therefore have densities whose first lower order terms differ by an explicit nonzero amount whenever their levels have different bounded prime factors.

math.NT↗

arXiv2026

Gaps for PLRS with zero coefficients

Zeckendorf proved that every positive integer has a unique decomposition as a sum of non-adjacent Fibonacci numbers. Among its extensions, we focus on Positive Linear Recurrence Sequences (PLRS): fixed depth recurrences with non-negative integral coefficients whose first and last terms are positive: $G_{n+1} = c_1 G_n + \dots + c_L G_{n-L+1}$ with $c_i$ non-negative integers and $c_1 c_L \ge 1$. We again find every integer can be written uniquely as an appropriate sum of terms of the sequence, and we can study the distribution of gaps between summands in decompositions. Bower et al. (2015) proved that if each $c_i \ge 1$ then the distribution of gaps in the generalized Zeckendorf decompositions of integers in $[G_n, G_{n+1})$ converges to a geometric decay for gaps at least two. What if some of the $c_i$'s are allowed to be zero? To date only the Kangaroo case, where there is a fixed $g$ such that $c_i = 1$ if $i \equiv 1 \bmod g$ and 0 otherwise, has been done and again geometric behavior emerges. Ben-Ari and Miller (2016) developed a probabilistic framework that also produce similar results, but their approach requires more advanced machinery. One of the goals of this paper is to show that the elementary methods used by Bower et al. (2015) can be adapted to this setting as well. We analyze the case $H_{n+1} = H_n + H_{n-p} + H_{n-p-q}$. For gaps at least $\max(p,q)$ we again prove geometric decay, with parameter the largest root of the characteristic polynomial associated with the recurrence. For the finitely many smaller gaps, under the restriction $p<q+2$, we prove that the limiting probabilities exist and can be computed using a transfer-matrix method, yielding results that depend on $p,q,$ and the gap size $g$. The case $p \ge q+2$ is not addressed by our small-gap argument.

math.NT↗

arXiv2026

The Bias Conjecture for One-Parameter, Non-Isotrivial Elliptic Curve Families

We investigate Miller's bias conjecture for general one-parameter families of elliptic curves over $\mathbb{Q}$ with non-constant $j$-invariants. Utilizing methods developed by Michel, we view the second moment as the sum of distinct cohomological components: a main $p^2$ term, a $p^{3/2}$ term arising from the first cohomology over the projective line, and terms of order $p$ or lower corresponding to singular fibers. By assuming the generalized Sato-Tate conjecture, we employ representation and motive theory to demonstrate that the coefficient of the $p^{3/2}$ term averages to zero in the limit. We then independently analyze the lower-order contributions without assuming Sato-Tate. By applying the Chebotarev density theorem and analyzing Galois representations, we show that the contributions to the order $p$ term from additive singular fibers also average to zero. Consequently, the only remaining average contribution to the $p$ term is $-B_pp$, where $B_p$ denotes the number of ``bad'' points where the discriminant vanishes. Because $B_p$ averages to a strictly positive value, the overall coefficient of the $p$ term averages to a strictly negative value in the limit, thereby establishing the bias conjecture for these families.

math.AG↗

arXiv2026

On digit-appending walks along linear recurrences

We study digit-appending walks in base $b\geq 2$ along integer linear recurrences with minimal characteristic polynomials having a unique simple dominant root $α$ that lies outside the unit circle. We prove that if $α$ and $b$ are multiplicatively independent, then no infinite walk can have uniformly bounded appended digit lengths. In contrast, we show that when the appended digit length can be unbounded, the same independence hypothesis implies the existence of infinite walks.

math.NT↗

arXiv2026

Product and Quotient Set Dominance in Finite Groups

A classic problem in additive number theory asks whether a given subset of integers has more sums or differences. This inquiry naturally extends to finite groups, where one compares the sizes of the product set $AA$ and the quotient set $AA^{-1}$ for a subset $A \subseteq G$. A finite group is considered product-dominant if it contains a higher proportion of subsets where $|AA| > |AA^{-1}|$, and quotient-dominant if the reverse is true. We establish statistical criteria to determine the dominance behavior of large finite groups by analyzing the moments of the random variable $d(A) := |AA| - |AA^{-1}|$. By translating product and quotient relationships into graph-theoretic constructions, we compute these moments through counting independent sets. Our main result demonstrates that for a growing sequence of finite groups $G_k$ with center size $Z_k$ and involution count $T_k$, the condition $Z_k = o(T_k)$ implies the groups are eventually product-dominant. Conversely, if $T_k \cdot \sqrt{3}^{R_k/Z_k} = o(Z_k)$ (where $R_k$ denotes the number of elements whose square is central), the groups are eventually quotient-dominant. Using these asymptotic conditions, we resolve the Miller--Vissuet conjecture by proving that all dihedral groups $D_n$ for $n \geq 3$ are product-dominant. Furthermore, we classify the dominance behavior of several major families of groups, proving that symmetric groups and non-abelian finite simple groups are product-dominant, while groups of odd order and certain abelian groups are quotient-dominant. Finally, for groups that fall outside our asymptotic conditions, we derive an explicit formula for the first moment utilizing the irreducible characters and conjugacy classes of the group.

math.NT↗

arXiv2026

Generating and generalizing MSTD sets through Markov processes

The classical More Sums Than Differences (MSTD) problem studies finite sets $A\subset\{0,1,\ldots,n\}$ for which $|A+A|>|A-A|$, where $A+A=\{a_1+a_2:a_1,a_2\in A\}$ and $A-A=\{a_1-a_2:a_1,a_2\in A\}$. As addition is commutative and subtraction is not, it was conjectured that as $n\to\infty$ almost all subsets $A$ chosen uniformly from the power set of $\{0,1,\ldots,n\}$ are difference-dominated, and it was thus a surprise when Martin and O'Bryant proved a positive percentage of sets are sum-dominant. We greatly generalize this model by introducing a Markov-chain framework, where the classical MSTD model is now just a special case. Let $(X_i)_{i=0}^n$ be a stationary two-state Markov chain on $\{0,1\}$ with transition probabilities $P(0,0)=p$ and $P(1,1)=q$, where $p,q\in(0,1)$. We include $i$ in $A$ exactly when $X_i=1$, and define $A=\{i\in\{0,\ldots,n\}:X_i=1\}$. The usual independent Bernoulli model is recovered when consecutive inclusion decisions are independent, equivalently when $p=1-q$. In particular, the uniformly random subset model corresponds to $p=q=1/2$. Using the fringe-middle method from the MSTD literature, we show that the middle sums and differences are filled with high probability, so the comparison between $|A+A|$ and $|A-A|$ is again governed by endpoint fringes. By fringe manipulation, we prove that the probabilities of sum-dominant, difference-dominant, and balanced sets tend to strictly positive limits as $n\to\infty$. We also give numerical estimates of these three probabilities for finite $n$ over a range of values of $p$ and $q$. Through combinatorial methods, we find a closed-form expression for $\mathbb{E}[|A-A|-|A+A|]$ as $n\to\infty$.

math.NT↗

arXiv2026

Why Does the Slope Definition of $e$ Work?: A Self-Contained Exploration After Real Analysis

Euler's number $e$ is often introduced through compound interest, an infinite series, an area, or the slope of an exponential function. The last viewpoint is especially attractive: choose the base $b\in (2,3)$ so that the graph of $y=b^x$ has tangent slope $1$ at $(0,1)$. However, a curious student may raise several questions that a brief introduction necessarily leaves implicit. What does $b^x$ mean when $x$ is irrational? Why is $b^x$ differentiable at $x = 0$? How can one show that the slopes corresponding to bases $2$ and $3$ lie on opposite sides of slope $1$? Finally, why is there a unique base whose slope is exactly $1$? The paper provides an answer using tools from a first course in real analysis, including limits, continuity, derivatives, and the Intermediate Value Theorem. Its primary audience is undergraduate students familiar with standard concepts from an introductory analysis course who wish to apply what they have learned to explore questions left open in earlier courses. A secondary audience is instructors who may use selected portions as supplementary material in an introduction-to-proofs course. The mathematical ingredients are classical; our aim is to organize them into a self-contained exposition and to make explicit the logical steps that are often compressed in standard treatments.

math.HO↗

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

On the Density of Low Lying Zeros of a Large Family of Automorphic $L$-functions

Under the generalized Riemann Hypothesis (GRH), Baluyot, Chandee, and Li nearly doubled the range in which the density of low lying zeros predicted by Katz and Sarnak is known to hold for a large family of automorphic $L$-functions with orthogonal symmetry. We generalize their main techniques to the study of higher centered moments of the one-level density of this family, leading to better results on the behavior near the central point. Numerous technical obstructions emerge that are not present in the one-level density. Averaging over the level of the forms and assuming GRH, we prove the density predicted by Katz and Sarnak holds for the $n$-th centered moments for test functions whose Fourier transform is compactly supported in $(-σ, σ)$ for $σ~=~\min\left\{3/2(n-1), 4/(2n-\mathbf{1}_{2\nmid n})\right\}$. For $n=3$, our results improve the previously best known $σ=2/3$ to $σ=3/4$. We also prove the two-level density agrees with the Katz-Sarnak density conjecture for test functions whose Fourier transform is compactly supported in $σ_1 = 3/2$ and $σ_2 = 5/6$, respectively, extending the previous best known sum of supports $σ_1 + σ_2 = 2$. This work is the first evidence of an interesting new phenomenon: by taking different test functions, we are able to extend the range in which the Katz-Sarnak density predictions hold. The techniques we develop can be applied to understanding quantities related to this family containing sums over multiple primes.

math.NT↗

arXiv2026

Benford behavior resulting from stick and box fragmentation processes

Benford's law is the statement that in many real world data sets, the probability of having digit $d$ in base $B$ as the first digit is \log_{B}\!\left(\frac{d+1}{d}\right) for all $1 \leq d \leq B$. We sometimes refer to this as weak Benford behavior, and we say that a data set satisfies strong Benford behavior in base $B$ if the probability of having significand at most $s$ is \log_{B}\!\left(s\right) for all $1 \leq s < B$, . We examine Benford behaviors in two different probabilistic models: stick and box fragmentation models. Building on the work arXiv:1309.5603 on the single proportion stick fragmentation model, we employ combinatorial identities on multinomial coefficients to reduce the multi-proportion stick fragmentation model to the single proportion model. We then provide a necessary and sufficient condition for the lengths of the stick fragments to converge to strong Benford behavior along with a quantification of the discrepancy from uniform distribution on $[0,1]$ in terms of irrationality exponent. Then we answer a conjecture of arXiv:2304.08335 on the high-dimensional box fragmentation model. Using tools from Fourier analysis and order statistics, we prove that under some mild conditions, faces of any arbitrary dimension of the box have total volume converging to strong Benford behavior.

math.PR↗