arXiv subjects
Publications and source records attributed to Wooyeon Kim.
We give a new proof of the quantitative Oppenheim theorem in signature $(2,2)$, originally proved by Eskin-Margulis-Mozes, by recasting the problem as one about determinant values on lattices in $\operatorname{M}_2(\mathbb R)$.
We study the distribution of determinant values on lattices in $\operatorname{M}_n(\mathbb R)$ for $n\ge 2$. Let $\Lambda<\operatorname{M}_n(\mathbb R)$ be a lattice whose elements have algebraic entries. We prove that if $\det (\Lambda)$ is not contained in a scalar multiple of $\mathbb Z$, then for every $a 0$ depends only on $n$. Under an additional hypothesis on the isotropic subspaces associated with $\Lambda$, which is automatic for $n=2,3$ and is satisfied when $n\geq 4$, by every lattice spanned by scalar multiples of the elementary matrices, we also obtain an asymptotic formula for the singular lattice points. Both conclusions extend to the broader class of Diophantine lattices under the corresponding hypotheses. For $n=2$, our theorem recovers the Eskin-Margulis-Mozes theorem on the quantitative Oppenheim problem for quadratic forms of signature $(2,2)$; more generally, our results may be viewed as higher-degree analogues of that theorem.
We prove a local-global principle for primitive representations of binary quadratic forms by quaternary quadratic forms. Our method is a variant of Linnik's ergodic method showing density for certain homogenous toral sets. The central ingredient is a measure classification result of Einsiedler and Lindenstrauss for actions of rank two diagonalizable groups on quotients of products of $\mathrm{SL}_2$. This rigidity result together with an application of the Siegel mass formula reduces the density problem to a counting problem on a certain affine variety. We solve that counting problem using the determinant method of Bombieri-Pila and Heath-Brown.
Berry and Tabor conjectured in 1977 that spectra of generic integrable quantum systems have the same local statistics as a Poisson point process. We verify their conjecture in the case of the two-point spectral density for a quantum particle in a three-dimensional box, subject to a Diophantine condition on the domain's proportions. A permissible choice of width, height and depth is for example $1,2^{1/3},2^{-1/3}$. This extends previous work of Eskin, Margulis and Mozes (Annals of Math., 2005) in dimension two, where the problem reduces to the quantitative Oppenheim conjecture for quadratic forms of signature $(2,2)$. The difficulty in three and higher dimensions is that we need to consider the distribution of indefinite forms in shrinking rather than fixed intervals, which we are able to resolve for special diagonal forms of signature $(3,3)$ in various scalings, including a rate of convergence. A key step of our approach is to represent the relevant counting problem as an average of a theta function on $\mathrm{SL}(2,\mathbb{Z})^3\backslash\mathrm{SL}(2,\mathbb{R})^3$ over an expanding family of one-parameter unipotent orbits. The asymptotic behaviour of these unipotent averages follows from Ratner's measure classification theorem and subtle escape of mass estimates.
We prove a local-global principle for representations of binary by quaternary quadratic forms. One of the main ingredients is a recent measure rigidity result of Einsiedler and Lindenstrauss for diagonalizable actions on quotients of products of $\mathrm{SL}_2$'s. Based on this, it suffices to show that limits of the uniform measures on the associated rank one adelic toral packets have more entropy than one half of the maximal entropy. The latter is proved using the Siegel mass formula and the determinant method as developed by Bombieri and Pila as well as Heath-Brown.
For $d\ge 3$ we first show that the Hausdorff dimension of the set of $A$-divergent on average points in the $(d-1)$-dimensional closed horosphere in the space of $d$-dimensional Euclidean lattices, where $A$ is the group of positive diagonal matrices, is at most $\frac{d-1}{2}$. In particular, this upper bound is sharp for $d=3$. We apply this to compute the Hausdorff dimension of the set of exceptions to the inhomogeneous uniform version of Littlewood conjecture. We say that a pair $(\xi_1,\xi_2)\in\mathbb{R}^2$ satisfies the inhomogeneous Littlewood conjecture if $$\liminf_{q\to\infty}q\|q\xi_1-\theta_1\|_{\mathbb{Z}}\|q\xi_2-\theta_2\|_{\mathbb{Z}}=0$$ for all $(\theta_1,\theta_2)\in\mathbb{R}^2$, where $\|\cdot\|_\mathbb{Z}$ denotes the distance to the nearest integer. We prove that the Hausdorff dimension of the set of pairs $(\xi_1,\xi_2)\in\mathbb{R}^2$ not satisfying the inhomogeneous Littlewood conjecture is $1$, which is equal to the Hausdorff dimension of the conjectural set of exceptions.
In this paper, we prove a quantitative version of the Oppenheim conjecture for indefinite ternary quadratic forms: for any indefinite irrational ternary quadratic form $Q$ that is not extremely well approxiable by rational forms, and for $a 0$ depends only on $Q$, and the term $\mathsf{I}_{Q}(a,b)T$ accounts for the contribution from rational isotropic lines and degenerate planes. The main technical ingredient is a uniform bound for the $\lambda$-moment of the Margulis $\alpha$-function along expanding translates of a unipotent orbit in $\operatorname{SL}_3(\mathbb{R})/\operatorname{SL}_3(\mathbb{Z})$, for some $\lambda>1$. To establish this, we introduce a new height function $\widetilde{\alpha}$ on the space of lattices, which captures the failure of the classical Margulis inequality. This moment bound implies equidistribution of such translates with respect to a class of unbounded test functions, including the Siegel transform.
We prove effective density of random walks on homogeneous spaces, assuming that the underlying measure is supported on matrices generating a dense subgroup and having algebraic entries. The main novelty is an argument passing from high dimension to effective equidistribution in the setting of random walks on homogeneous spaces, exploiting spectral gap of the associated convolution operator.
We prove a polynomially effective equidistribution result for expanding translates in the space of $d$-dimensional affine lattices for any $d\ge 2$.
We prove the convergence of moments of the number of directions of affine lattice vectors that fall into a small disc, under natural Diophantine conditions on the shift. Furthermore, we show that the pair correlation function is Poissonian for any irrational shift in dimension 3 and higher, including well-approximable vectors. Convergence in distribution was already proved in the work of Str\"ombergsson and the second author, and the principal step in the extension to convergence of moments is an escape of mass estimate for averages over embedded $\operatorname{SL}(d,\mathbb{R})$-horospheres in the space of affine lattices.
For given $ε>0$ and $b\in\mathbb{R}^m$, we say that a real $m\times n$ matrix $A$ is $ε$-badly approximable for the target $b$ if $$\liminf_{q\in\mathbb{Z}^n, \|q\|\to\infty} \|q\|^n \langle Aq-b \rangle^m \geq ε,$$ where $\langle \cdot \rangle$ denotes the distance from the nearest integral point. In this article, we obtain upper bounds for the Hausdorff dimensions of the set of $ε$-badly approximable matrices for fixed target $b$ and the set of $ε$-badly approximable targets for fixed matrix $A$. Moreover, we give an equivalent Diophantine condition of $A$ for which the set of $ε$-badly approximable targets for fixed $A$ has full Hausdorff dimension for some $ε>0$. The upper bounds are established by effectivizing entropy rigidity in homogeneous dynamics, which is of independent interest. For the $A$-fixed case, our method also works for the weighted setting where the supremum norms are replaced by certain weighted quasinorms.
For a decreasing real valued function $ψ$, a pair $(A,\mathbf{b})$ of a real $m\times n$ matrix $A$ and $\mathbf{b}\in\mathbb{R}^m$ is said to be $ψ$-Dirichlet improvable if the system $$\|A\mathbf{q}+\mathbf{b}-\mathbf{p}\|^m < ψ(T)\quad\text{and}\quad\|\mathbf{q}\|^n < T$$ has a solution $\mathbf{p}\in\mathbb{Z}^m$, $\mathbf{q}\in\mathbb{Z}^n$ for all sufficiently large $T$, where $\|\cdot\|$ denotes the supremum norm. Kleinbock and Wadleigh (2019) established an integrability criterion for the Lebesgue measure of the $ψ$-Dirichlet non-improvable set. In this paper, we prove a similar criterion for the Hausdorff measure of the $ψ$-Dirichlet non-improvable set. Also, we extend this result to the singly metric case that $\mathbf{b}$ is fixed. As an application, we compute the Hausdorff dimension of the set of pairs $(A,\mathbf{b})$ with uniform Diophantine exponents $\widehat{w}(A,\mathbf{b})\leq w$.
We prove that for all $b$, the Hausdorff dimension of the set of $m \times n$ matrices $ε$-badly approximable for the target $b$ is not full. The doubly metric case follows. It was known that for almost every matrix $A$, the Hausdorff dimension of the set $Bad_A(ε)$ of $ε$-badly approximable target $b$ is not full, and that for real numbers $α$, $\dim_H Bad_α(ε)=1$ if and only if $α$ is singular on average. We show that if $\dim_H Bad_A(ε)=m$, then $A$ is singular on average.
Volume entropy is an important invariant of metric graphs as well as Riemannian manifolds. In this note, we calculate the change of volume entropy when an edge is added to a metric graph. Using the first result, we investigate the change of volume entropy when a vertex and edges around it are added. In the second part, we estimate the value of the volume entropy which can be used to suggest an algorithm of calculating the persistent volume entropy of graphs.