arXiv subjects

Samantha Fairchild

Publications and source records attributed to Samantha Fairchild.

9 recordsLinked to original sources

arXiv2026

Densities of arithmetic Hecke triangle group orbits

We give new proofs computing the asymptotic densities for orbits of the arithmetic Hecke triangle groups $Γ_q$ when $q=4$ and $q=6$. We use elementary number theory techniques along with basic properties of the Möbius function and Riemann zeta function with additional congruence conditions. This note actually came about by observing that, in the $q=4$ case, the underlying congruence condition partitions the set of coprime integer pairs into three classes of equal density.

math.NT↗

arXiv2025

Mean value theorems for the S-arithmetic primitive Siegel transforms

We develop the theory and properties of primitive unimodular $S$-arithmetic lattices in $\mathbb{Q}_S^d$ by giving integral formulas in the spirit of Siegel's primitive mean value formula and Rogers' and Schmidt's second moment formulas. When $d=2$, unlike in the real case, functions arising from the $S$-primitive Siegel transform are unbounded, requiring a careful analysis to establish their integrability. We then use mean value and second moment formulas in three applications. First, we obtain quantitative estimates for counting primitive $S$-arithmetic lattice points. We next establish a quantitative Khintchine--Groshev theorem, which, in the real case, involves counting primitive integer points in $\mathbb{Z}^d$ subject to congruence conditions. Finally, we derive an $S$-arithmetic logarithm law for unipotent flows in the spirit of Athreya--Margulis. These applications follow the spirit of the real case, but require new technical aspects of the proofs, particularly when $d=2$.

math.NT↗

arXiv2024

Shrinking rates of horizontal gaps for generic translation surfaces

A translation surface is given by polygons in the plane, with sides identified by translations to create a closed Riemann surface with a flat structure away from finitely many singular points. Understanding geodesic flow on a surface involves understanding saddle connections. Saddle connections are the geodesics starting and ending at these singular points and are associated to a discrete subset of the plane. To measure the behavior of saddle connections of length at most $R$, we obtain precise decay rates as $R\to \infty$ for the difference in angle between two almost horizontal saddle connections.

math.DS↗

arXiv2024

Pairs in discrete lattice orbits with applications to Veech surfaces

Let $Λ_1$, $Λ_2$ be two discrete orbits under the linear action of a lattice $Γ<\mathrm{SL}_2(\mathbb{R})$ on the Euclidean plane. We prove a Siegel$-$Veech-type integral formula for the averages $$ \sum_{\mathbf{x}\inΛ_1} \sum_{\mathbf{y}\inΛ_2} f(\mathbf{x}, \mathbf{y}) $$ from which we derive new results for the set $S_M$ of holonomy vectors of saddle connections of a Veech surface $M$. This includes an effective count for generic Borel sets with respect to linear transformations, and upper bounds on the number of pairs in $S_M$ with bounded determinant and on the number of pairs in $S_M$ with bounded distance. This last estimate is used in the appendix to prove that for almost every $(θ,ψ)\in S^1\times S^1$ the translations flows $F_θ^t$ and $F_ψ^t$ on any Veech surface $M$ are disjoint.

math.DS↗

arXiv2024

Crossing the transcendental divide: from Schottky groups to algebraic curves

Though the uniformization theorem guarantees an equivalence of Riemann surfaces and smooth algebraic curves, moving between analytic and algebraic representations is inherently transcendental. Our analytic curves identify pairs of circles in the complex plane via free groups of Möbius transformations called Schottky groups. We construct a family of non-hyperelliptic surfaces of genus $g\geq 3$ where we know the Riemann surface as well as properties of the canonical embedding, including a nontrivial symmetry group and a real structure with the maximal number of connected components (an $M$-curve). We then numerically approximate the algebraic curve and Riemann matrices underlying our family of Riemann surfaces.

math.GT↗

arXiv2023

Average degree of the essential variety

The essential variety is an algebraic subvariety of dimension $5$ in real projective space $\mathbb R\mathrm P^{8}$ which encodes the relative pose of two calibrated pinhole cameras. The $5$-point algorithm in computer vision computes the real points in the intersection of the essential variety with a linear space of codimension $5$. The degree of the essential variety is $10$, so this intersection consists of 10 complex points in general. We compute the expected number of real intersection points when the linear space is random. We focus on two probability distributions for linear spaces. The first distribution is invariant under the action of the orthogonal group $\mathrm{O}(9)$ acting on linear spaces in $\mathbb R\mathrm P^{8}$. In this case, the expected number of real intersection points is equal to $4$. The second distribution is motivated from computer vision and is defined by choosing 5 point correspondences in the image planes $\mathbb R\mathrm P^2\times \mathbb R\mathrm P^2$ uniformly at random. A Monte Carlo computation suggests that with high probability the expected value lies in the interval $(3.95 - 0.05,\ 3.95 + 0.05)$.

math.AG↗

arXiv2023

Crossing the transcendental divide: from translation surfaces to algebraic curves

We study constructing an algebraic curve from a Riemann surface given via a translation surface, which is a collection of finitely many polygons in the plane with sides identified by translation. We use the theory of discrete Riemann surfaces to give an algorithm for approximating the Jacobian variety of a translation surface whose polygon can be decomposed into squares. We first implement the algorithm in the case of $L$ shaped polygons where the algebraic curve is already known. The algorithm is also implemented in any genus for specific examples of Jenkins-Strebel representatives, a dense family of translation surfaces that, until now, lived squarely on the analytic side of the transcendental divide between Riemann surfaces and algebraic curves. Using Riemann theta functions, we give numerical experiments and resulting conjectures up to genus 5.

math.AG↗

arXiv2020

Families of well approximable measures

We provide an algorithm to approximate a finitely supported discrete measure $μ$ by a measure $ν_{N}$ corresponding to a set of $N$ points so that the total variation between $μ$ and $ν_N$ has an upper bound. As a consequence if $μ$ is a (finite or infinitely supported) discrete probability measure on $[0,1]^{d}$ with a sufficient decay rate on the weights of each point, then $μ$ can be approximated by $ν_N$ with total variation, and hence star-discrepancy, bounded above by $(\log N) N^{-1}$. Our result improves, in the discrete case, recent work by Aistleitner, Bilyk, and Nikolov who show that for any normalized Borel measure $μ$, there exist finite sets whose star-discrepancy with respect to $μ$ is at most $(\log N)^{d-\frac{1}{2}} N^{-1}$. Moreover we close a gap in the literature for discrepancy in the case $d=1$ showing both that Lebesgue is indeed the hardest measure to approximate by finite sets and also that all measures without discrete components have the same order of discrepancy as the Lebesgue measure.

math.NT↗

arXiv2019

The Abelian Sandpile Model on Fractal Graphs

We study the Abelian sandpile model (ASM), a process where grains of sand are placed on a graph's vertices. When the number of grains on a vertex is at least its degree, one grain is distributed to each neighboring vertex. This model has been shown to form fractal patterns on the integer lattice, and using these fractal patterns as motivation, we consider the model on graph approximations of post critically finite (p.c.f) fractals. We determine asymptotic behavior of the diameter of sites toppled and characterize graphs which exhibit a periodic number of grains with respect to the initial placement.

math.PR↗