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Samuel Everett

Publications and source records attributed to Samuel Everett.

11 recordsLinked to original sources
arXiv2026

Lonely Runner Relations

We study the Lonely Runner Conjecture (LRC), conceived by J\"org M. Wills in the 1960's: Given positive integers $n_1, n_2, \dots, n_k$, there exists a positive real number $t$ such that for all $1 \le j \le k$ the distance of $t \,n_j$ to the nearest integer is at least $\frac{ 1 }{ k+1 }$. We prove that for any counterexample or tight instance $\mathbf{n}$ of LRC, $\mathbf{m} \cdot \mathbf{n} = 0$ for some $\mathbf{m} \in \mathbb{Z}^k$ with $0 < \| \mathbf{m} \|_1 \le \min(2k+3, \ \frac{ k+1 }{ k-1 } \mathrm{flt}(k))$ where $\mathrm{flt}(k)$ denotes Khinchin's (1948) flatness constant limiting the lattice width of a $k$-dimensional convex body without interior integer points. In other words, potential counterexamples to LRC lie on a finite set of hyperplanes in the parameter space. Our proofs use Fourier analysis and a geometric reformulation of LRC, and our results generalize to the situation of shifted lonely runners of varying measures of loneliness. Our results imply and generalize a theorem of Czerwi\'nski (2012) that when we choose $\mathbf{n}$ at random then, with probability tending to 1, the measure of loneliness $\frac{1}{ k+1 }$ can be replaced by $\frac 1 2 - \epsilon$.

math.CO↗
arXiv2026

Random tensor isomorphism under orthogonal and unitary actions

We study the problem of testing whether two tensors in $\mathbb{R}^\ell\otimes \mathbb{R}^m\otimes \mathbb{R}^n$ are isomorphic under the natural action of orthogonal groups $\textbf{O}(\ell, \mathbb{R})\times\textbf{O}(m, \mathbb{R})\times\textbf{O}(n, \mathbb{R})$, as well as the corresponding question over $\mathbb{C}$ and unitary groups. These problems naturally arise in several areas, including graph and tensor isomorphism (Grochow--Qiao, SIAM J. Comp. '21), scaling algorithms for orbit closure intersections (Allen-Zhu--Garg--Li--Oliveira--Wigderson, STOC '18), and quantum information (Liu--Li--Li--Qiao, Phys. Rev. Lett. '12). We study average-case algorithms for orthogonal and unitary tensor isomorphism, with one random tensor where each entry is sampled uniformly independently from a sub-Gaussian distribution, and the other arbitrary. For the algorithm design, we develop algorithmic ideas from the higher-order singular value approach into polynomial-time exact (algebraic) and approximate (numerical) algorithms with rigorous average-case analyses. Following (Allen-Zhu--Garg--Li--Oliveira--Wigderson, STOC '18), we present an algorithm for a gapped version of the orbit distance approximation problem. For the average-case analysis, we work from recent progress in random matrix theory on eigenvalue repulsion of sub-Gaussian Wishart matrices (Christoffersen--Luh--O'Rourke--Shearer and Han, arXiv '25) by extending their results from side lengths of Wishart matrices linearly related to polynomially related.

cs.CC↗
arXiv2026

Correspondences in computational and dynamical complexity II: forcing complex reductions

An algebraic telic problem is a decision problem in $\textsf{NP}_\mathbb{R}$ formalizing finite-time reachability questions for one-dimensional dynamical systems. We prove that the existence of "natural" mapping reductions between algebraic telic problems coming from distinct dynamical systems implies the two dynamical systems exhibit similar behavior (in a precise sense). As a consequence, we obtain explicit barriers for algorithms solving algebraic telic problems coming from complex dynamical systems, such as those with positive topological entropy. For example, some telic problems cannot be decided by uniform arithmetic circuit families with only $+$ and $\times$ gates.

cs.CC↗
arXiv2026

Correspondences in computational and dynamical complexity I

We begin development of a method for studying dynamical systems using concepts from computational complexity theory. We associate families of decision problems, called telic problems, to dynamical systems of a certain class. These decision problems formalize finite-time reachability questions for the dynamics with respect to natural coarse-grainings of state space. Our main result shows that complexity-theoretic lower bounds have dynamical consequences: if a system admits a telic problem for which every decider runs in time $2^{\Omega(n)}$, then it must have positive topological entropy. This result and others lead to methods for classifying dynamical systems through proving bounds on the runtime of algorithms solving their associated telic problems, or by constructing polynomial-time reductions between telic problems coming from distinct dynamical systems.

math.DS↗
arXiv2024

Deciding subspace reachability problems with application to Skolem's Problem

The higher-dimensional version of Kannan and Lipton's Orbit Problem asks whether it is decidable if a target subspace can be reached from a starting point under repeated application of a linear transformation. Similarly, the continuous analog of the Orbit Problem asks if a flow induced by a linear system of differential equations ever reaches some specified subspace. The decidability of both problems remains open, and in fact the problems generalize the discrete and continuous versions of Skolem's Problem. The object of this paper is to communicate a geometric perspective of the discrete and continuous Orbit Problems, alternate to the traditional and highly technical algebraic and number-theoretic approaches to the problem. We derive a simple decision procedure capable of deciding a certain class of instances of the Orbit Problem, and, as an application, we obtain alternate proofs to a number of results using elementary geometric arguments.

cs.LO↗
arXiv2024

A piecewise contractive map on triangles

We study the dynamics of a piecewise map defined on the set of three pairwise nonparallel, nonconcurrent lines in $\mathbb{R}^2$. The geometric map of study may be analogized to the billiard map with a different reflection rule so that each iteration is a contraction over the space, thereby providing asymptotic behavior of interest. Our study emphasizes the behavior of periodic orbits generated by the map, with description of their geometry and bifurcation behavior. We establish that for any initial point in the space, the orbit will converge to a fixed point or periodic orbit, and we demonstrate that there exists an infinite variety of periodic orbits the orbits may converge to, dependent on the parameters of the underlying space.

math.DS↗
arXiv2024

On the use of dynamical systems in cryptography

Ever since the link between nonlinear science and cryptography became apparent, the problem of applying chaotic dynamics to the construction of cryptographic systems has gained a broad audience and has been the subject of thousands of papers. Yet, the field has not found its place in mainstream cryptography, largely due to persistent weaknesses in the presented systems. The goal of this paper is to help remedy this problem in two ways. The first is by providing a new algorithm that can be used to attack -- and hence test the security of -- stream ciphers based on the iteration of a chaotic map of the interval. The second is to cast discrete dynamical systems problems in a modern cryptographic and complexity theoretic language, so that researchers working in chaos-based cryptography can begin designing cryptographic protocols that have a better chance of meeting the extreme standards of modern cryptography.

cs.CR↗
arXiv2021

Computing Periodic Points on Veech Surfaces

A non-square-tiled Veech surface has finitely many periodic points, i.e., points with finite orbit under the affine automorphism group. We present an algorithm that inputs a non-square-tiled Veech surface and outputs its set of periodic points. We apply our algorithm to Prym eigenforms in the minimal stratum in genus 3, proving that in low discriminant these surfaces do not have periodic points, except for the fixed points of the Prym involution.

math.DS↗
arXiv2021

A geometric dynamical system with relation to billiards

We introduce a geometric dynamical system where iteration is defined as a cycling composition of different maps acting on a space composed of three or more lines in $\mathbb{R}^2$. This system is motivated by the dynamics of iterated function systems, as well as billiards with modified reflection laws. We provide conditions under which this dynamical system generates periodic orbits, and use this result to prove the existence of closed nonsmooth curves over $\mathbb{R}^2$ which satisfy particular structural constraints with respect to a space of intersecting lines in the plane.

math.DS↗
arXiv2021

Long and Short Periodic Billiard Trajectories in the Regular Pentagon

In any periodic direction on the regular pentagon billiard table, there exists two combinatorially different billiard paths, with one longer than the other. For each periodic direction, McMullen asked if one could determine whether the periodic trajectory through a given point is long, short, or a saddle connection. In this paper we present an algorithm resolving this question for trajectories emanating from the midpoints of the pentagon.

math.DS↗