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Jimmy Tseng

Publications and source records attributed to Jimmy Tseng.

At least 19 recordsLinked to original sources
arXiv2026

$\delta$-Badly approximable numbers and ubiquitously losing sets

We consider a natural filtration $\boldsymbol{\operatorname{Bad}}(\delta) \subset \boldsymbol{\operatorname{Bad}}(\delta')$ for $\delta \geq \delta'>0$ on the set of badly approximable numbers to complement the filtration of the well approximable numbers by the $\tau$-well approximable numbers. We show that the set $\boldsymbol{\operatorname{Bad}}(\delta)$ is a $(1/3, 18 \delta)$-winning set and give a lower bound on its Hausdorff dimension. We introduce the notion of $(\alpha, \beta)$-$\textit{ubiquitously losing sets}$ to the theory of Schmidt games, give an upper bound on the Hausdorff dimension of an $(\alpha, \beta)$-ubiquitously losing set that is strictly less than full Hausdorff dimension, show that $\boldsymbol{\operatorname{Bad}}(\delta)$ is a $(1/2, 18/\delta)$-ubiquitously losing set, and give an upper bound on the Hausdorff dimension of $\boldsymbol{\operatorname{Bad}}(\delta)$ that is strictly less than one. Combined with a finite intersection property and a bilipschitz transfer property, we obtain results for finite intersections of translates of $\boldsymbol{\operatorname{Bad}}(\delta)$.

math.NT↗
arXiv2023

An asymptotic for the K-Bessel function using the saddle-point method

Using the saddle-point method, we compute an asymptotic, as $y \rightarrow \infty$, for the $K$-Bessel function $K_{r + i t}(y)$ with positive, real argument $y$ and of large complex order $r+it$ where $r$ is bounded and $t = y \sin \theta$ for a fixed parameter $0\leq \theta\leq \pi/2$ or $t= y \cosh \mu$ for a fixed parameter $\mu>0$. Our method gives an illustrative proof, using elementary tools, of this known result and explains how these asymptotics come about. As part of our proof, we prove a new result, namely a novel integral representation for $K_{r + i t}(y)$ in the case $t= y \cosh \mu$. This integral representation involves only one saddle point.

math.CA↗
arXiv2022

Shrinking target horospherical equidistribution via translated Farey sequences

For a certain diagonal flow on $\operatorname{SL}(d, \mathbb{Z}) \backslash \operatorname{SL}(d, \mathbb{R})$ where $d \geq 2$, we show that any bounded subset (with measure zero boundary) of the horosphere or a translated horosphere equidistributes, under a suitable normalization, on a target shrinking into the cusp. This type of equidistribution is shrinking target horospherical equidistribution (STHE), and we show STHE for several types of shrinking targets. Our STHE results extend known results for $d=2$ and $\mathcal{L} \backslash \operatorname{PSL}(2, \mathbb{R})$ where $\mathcal{L}$ is any cofinite Fuchsian group with at least one cusp. The two key tools needed to prove our STHE results for the horosphere are a renormalization technique and Marklof's result on the equidistribution of the Farey sequence on distinguished sections. For our STHE results for translated horospheres, we introduce translated Farey sequences, develop some of their geometric and dynamical properties, generalize Marklof's result by proving the equidistribution of translated Farey sequences for the same distinguished sections, and use this equidistribution of translated Farey sequences along with the renormalization technique to prove our STHE results for translated horospheres.

math.DS↗
arXiv2021

Shrinking target equidistribution of horocycles in cusps

Consider a shrinking neighborhood of a cusp of the unit tangent bundle of a noncompact hyperbolic surface of finite area, and let the neighborhood shrink into the cusp at a rate of $T^{-1}$ as $T \rightarrow \infty$. We show that a closed horocycle whose length $\ell$ goes to infinity or even a segment of that horocycle becomes equidistributed on the shrinking neighborhood when normalized by the rate $T^{-1}$ provided that $T/\ell \rightarrow 0$ and, for any $\delta>0$, the segment remains larger than $\max\left\{T^{-1/6},\left(T/\ell\right)^{1/2}\right\}\left(T/\ell\right)^{-\delta}$. We also have an effective result for a smaller range of rates of growth of $T$ and $\ell$. Finally, a number-theoretic identity involving the Euler totient function follows from our technique.

math.DS↗
arXiv2020

Eisenstein series and an asymptotic for the $K$-Bessel function

We produce an estimate for the $K$-Bessel function $K_{r + i t}(y)$ with positive, real argument $y$ and of large complex order $r+it$ where $r$ is bounded and $t = y \sin θ$ for a fixed parameter $0\leq θ\leq π/2$ or $t= y \cosh μ$ for a fixed parameter $μ>0$. In particular, we compute the dominant term of the asymptotic expansion of $K_{r + i t}(y)$ as $y \rightarrow \infty$. When $t$ and $y$ are close (or equal), we also give a uniform estimate. As an application of these estimates, we give bounds on the weight-zero (real-analytic) Eisenstein series $E_0^{(j)}(z, r+it)$ for each inequivalent cusp $κ_j$ when $1/2 \leq r \leq 3/2$.

math.NT↗
arXiv2016

Ergodic Theory and Diophantine approximation for translation surfaces and linear forms

We derive results on the distribution of directions of saddle connections on translation surfaces using only the Birkhoff ergodic theorem applied to the geodesic flow on the moduli space of translation surfaces. Our techniques, together with an approximation argument, also give an alternative proof of a weak version of a classical theorem in multi-dimensional Diophantine approximation due to W. Schmidt \cite{SchmidtMetrical, SchmidtMetrical2}. The approximation argument allows us to deduce the Birkhoff genericity of almost all lattices in a certain submanifold of the space of unimodular lattices from the Birkhoff genericity of almost all lattices in the whole space and similarly for the space of affine unimodular lattices.

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arXiv2016

Nondense orbits for Anosov diffeomorphisms of the $2$-torus

Let $λ$ denote the probability Lebesgue measure on ${\mathbb T}^2$. For any $C^2$-Anosov diffeomorphism of the $2$-torus preserving $λ$ with measure-theoretic entropy equal to topological entropy, we show that the set of points with nondense orbits is hyperplane absolute winning (HAW). This generalizes the result in~\cite[Theorem~1.4]{T4} for $C^2$-expanding maps of the circle.

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arXiv2015

Simultaneous dense and nondense orbits for toral diffeomorphisms

We show that, for pairs of hyperbolic toral automorphisms on the $2$-torus, the points with dense forward orbits under one map and nondense forward orbits under the other is a dense, uncountable set. The pair of maps can be noncommuting. We also show the same for pairs of $C^2$-Anosov diffeomorphisms on the $2$-torus. (The pairs must satisfy slight constraints.) Our main tools are the Baire Category theorem and a geometric construction that allows us to give a geometric characterization of the fractal that is the set of points with forward orbits that miss a certain open set.

math.DS↗
arXiv2015

Simultaneous dense and nondense orbits and the space of lattices

We show that set of points nondense under the $\times n$-map on the circle and dense for the geodesic flow under the induced map on the circle corresponding to the expanding horospherical subgroup has full Haudorff dimension. We also show the analogous result for toral automorphisms on the $2$-torus and a diagonal flow. Our results can be interpreted in number-theoretic terms: the set of well approximable numbers that are nondense under the $\times n$-map has full Hausdorff dimension. Similarly, the set of well approximable $2$-vectors that are nondense under a hyperbolic toral automorphism has full Hausdorff dimension. Our result for numbers is the counterpart to a classical result of Kaufmann and gives a comprehensive understanding.

math.DS↗
arXiv2014

Spiraling of approximations and spherical averages of Siegel transforms

We consider the question of how approximations satisfying Dirichlet's theorem spiral around vectors in $\mathbb{R}^d$. We give pointwise almost everywhere results (using only the Birkhoff ergodic theorem on the space of lattices). In addition, we show that for $\textit{every}$ unimodular lattice, on average, the directions of approximates spiral in a uniformly distributed fashion on the $d-1$ dimensional unit sphere. For this second result, we adapt a very recent proof of Marklof and Strömbergsson \cite{MS3} to show a spherical average result for Siegel transforms on $\operatorname{SL}_{d+1}(\mathbb{R})/\operatorname{SL}_{d+1}(\mathbb{Z})$. Our techniques are elementary. Results like this date back to the work of Eskin-Margulis-Mozes \cite{EMM} and Kleinbock-Margulis \cite{KM} and have wide-ranging applications. We also explicitly construct examples in which the directions are not uniformly distributed.

math.NT↗
arXiv2014

Simultaneous dense and nondense orbits for commuting maps

We show that, for two commuting automorphisms of the torus and for two elements of the Cartan action on compact higher rank homogeneous spaces, many points have drastically different orbit structures for the two maps. Specifically, using measure rigidity, we show that the set of points that have dense orbit under one map and nondense orbit under the second has full Hausdorff dimension.

math.DS↗
arXiv2012

Bounded Lüroth expansions: applying Schmidt games where infinite distortion exists

We show that the set of numbers with bounded Lüroth expansions (or bounded Lüroth series) is winning and strong winning. From either winning property, it immediately follows that the set is dense, has full Hausdorff dimension, and satisfies a countable intersection property. Our result matches the well-known analogous result for bounded continued fraction expansions or, equivalently, badly approximable numbers. We note that Lüroth expansions have a countably infinite Markov partition, which leads to the notion of infinite distortion (in the sense of Markov partitions).

math.NT↗
arXiv2011

Badly approximable vectors on rational quadratic varieties

Approximation in this paper is of vectors on the unit $d$-cube by the projection of integer lattice points onto the same cube. We define badly approximable vectors on a rational quadratic variety and show that sets of these vectors, which are (naturally) indexed by $m \in \QQ$, are winning and strong winning in the sense of Schmidt games. From the winning property, it follows that these sets have full Hausdorff dimension and, moreover, so does their intersection. In most cases, these sets are known to be null sets.

math.NT↗
arXiv2010

Remarks on shrinking target properties

This paper defines and describes a few (related) notions of shrinking target property. We show that simultaneous expanding circle maps have a certain shrinking target property, but that circle homeomorphisms and isometries of complete, separable metric spaces do not have any shrinking target property.

math.DS↗
arXiv2009

Badly approximable systems of affine forms, fractals, and Schmidt games

A badly approximable system of affine forms is determined by a matrix and a vector. We show Kleinbock's conjecture for badly approximable systems of affine forms: for any fixed vector, the set of badly approximable systems of affine forms is winning (in the sense of Schmidt games) even when restricted to a fractal (from a certain large class of fractals). In addition, we consider fixing the matrix instead of the vector where an analog statement holds.

math.DS↗
arXiv2008

Badly approximable affine forms and Schmidt games

For any real number \t, the set of all real numbers x for which there exists a constant c(x) > 0 such that \inf_{p \in \ZZ} |\t q - x - p| \geq c(x)/|q| for all q in \ZZ {0} is an 1/8-winning set.

math.NT↗
arXiv2008

Schmidt games and Markov partitions

Let T be a C^2-expanding self-map of a compact, connected, smooth, Riemannian manifold M. We correct a minor gap in the proof of a theorem from the literature: the set of points whose forward orbits are nondense has full Hausdorff dimension. Our correction allows us to strengthen the theorem. Combining the correction with Schmidt games, we generalize the theorem in dimension one: given a point x in M, the set of points whose forward orbit closures miss x is a winning set.

math.DS↗