arXiv subjects

Sunrose T. Shrestha

Publications and source records attributed to Sunrose T. Shrestha.

5 recordsLinked to original sources

arXiv2026

Complexity of Billiards in Polygons Associated to Hyperbolic $(p,q)$-Tilings

The complexity of the billiard language of regular polygons in the hyperbolic plane with $p$ sides and $2π/q$ internal angles is known to grow exponentially and the exponential growth rate is known to equal the topological entropy of the billiard system. In this paper we compute these exponential growth rates explicitly when $q$ is even and give bounds when $q$ is odd. Additionally, for the $q$ even case, we give complete grammar rules that establish when a word (finite, infinite or bi-infinite) in $p$ letters is realized by a billiard path. This latter result is roughly stated and not rigorously proved in a paper of Giannoni and Ullmo (1995). In this paper, we provide a precise statement and a complete proof using new methods relating to minimal tiling paths.

math.DS↗

arXiv2026

Straight-line trajectories on the Mucube

The dynamics of straight line flows on compact half-translation surfaces (surfaces formed by gluing Euclidean polygons edge-to-edge via translations possibly composed with rotation by $π$) has been widely studied due to their connections to polygonal billiards and Teichmüller theory. However, much less is known when the underlying surface is non-compact or infinite type. In this paper, we consider the straight line flow of the Mucube -- an infinite $\mathbb{Z}^3$-periodic half-translation square-tiled surface -- first written about by Coxeter and Petrie and more recently studied by Athreya--Lee and Gutiérrez-Romo--Lee--Sánchez. We give a geometric description of the flow's periodic and drift orbits in terms of the Mucube's rigid symmetries, and we give a complete characterization of the set of directions in which the straight line flow is periodic on the Mucube -- first in terms of a genus one quotient and second in terms of an infinitely generated subgroup of $\mathrm{SL}_2(\mathbb{Z})$. We use the latter characterization to obtain the Veech group (i.e. group of derivatives of affine diffeomorphisms) of the Mucube. Finally, we prove density of the sets of periodic and ergodic directions.

math.DS↗

arXiv2021

Asymptotic identities for additive convolutions of sums of divisors

In a 1916 paper, Ramanujan studied the additive convolution $S_{a, b}(n)$ of sum-of-divisors functions $σ_a(n)$ and $σ_b(n)$, and proved an asymptotic formula for it when $a$ and $b$ are positive odd integers. He also conjectured that his asymptotic formula should hold for all positive real $a$ and $b$. Ramanujan's conjecture was subsequently proved by Ingham, and then by Halberstam with a power saving error term. In this paper, we give a new proof of Ramanujan's conjecture that obtains lower order terms in the asymptotics for most ranges of the parameters. We also describe a connection to a counting problem in geometric topology that was studied in the second author's thesis and which served as our initial motivation in studying this sum.

math.NT↗

arXiv2019

Statistics of Square-tiled Surfaces: Symmetry and Short Loops

Square-tiled surfaces are a class of translation surfaces that are of particular interest in geometry and dynamics because, as covers of the square torus, they share some of its simplicity and structure. In this paper, we study counting problems that result from focusing on properties of the square torus one by one. After drawing insights from experimental evidence, we consider the implications between these properties and their frequency in each stratum of translation surfaces.

math.GT↗

arXiv2018

Counting Formulae for Square-tiled Surfaces in Genus Two

Square-tiled surfaces can be classified by their number of squares and their cylinder diagrams (also called realizable separatrix diagrams). For the case of $n$ squares and two cone points with angle $4 π$ each, we set up and parametrize the classification into four diagrams. Our main result is to provide formulae for enumeration of square-tiled surfaces in these four diagrams, completing the detailed count for genus two. The formulae are in terms of various well-studied arithmetic functions, enabling us to give asymptotics for each diagram using a new calculation for additive convolutions of divisor functions that was recently derived by the author and collaborators. Interestingly, two of the four cylinder diagrams occur with asymptotic density 1/4, but the other diagrams occur with different (and irrational) densities.

math.GT↗