Related papers: High-low method and $p$-adic Furstenberg set over …

Let $p$ be a prime number. We study the slopes of $U_p$-eigenvalues on the subspace of modular forms that can be transferred to a definite quaternion algebra. We give a sharp lower bound of the corresponding Newton polygon. The computation…

We establish Euclidean-type lower bounds for the codimension-1 Hausdorff measure of sets that separate points in doubling and linearly locally contractible metric manifolds. This gives a quantitative topological isoperimetric inequality in…

Two subanalytic subsets of R^n are called s-equivalent at a common point P if the Hausdorff distance between their intersections with the sphere centered at P of radius r vanishes of order greater than s when r tends to 0. In this paper we…

In this paper, we study stability problem of anisotropic capillary hypersurfaces in an Euclidean half-space. We prove that any compact immersed anisotropic capillary constant anisotropic mean curvature hypersurface in the half-space is…

On a family of arithmetic hyperbolic 3-manifolds of squarefree level, we prove an upper bound for the sup-norm of Hecke-Maass cusp forms, with a power saving over the local geometric bound simultaneously in the Laplacian eigenvalue and the…

We extend results by B. Hasselblatt, J. Schmeling in \emph{Dimension product structure of hyperbolic sets} (2004), and by the third author and K. Simon in \emph{Hausdorff and packing measures for solenoids} (2003), for $C^{1+\varepsilon}$…

We give a new proof of a theorem of Kleiner-Leeb: that any quasi-isometrically embedded Euclidean space in a product of symmetric spaces and Euclidean buildings is contained in a metric neighborhood of finitely many flats, as long as the…

Given a weight vector $\tau=(\tau_{1}, \dots, \tau_{n}) \in \mathbb{R}^{n}_{+}$ with each $\tau_{i}$ bounded by certain constraints, we obtain a lower bound for the Hausdorff dimension of the set of $\tau$-approximable points points over a…

Let $f$ be a newform of prime level $p$ with any central character $\chi\, (\bmod\, p)$, and let $g$ be a fixed cusp form or Eisenstein series for $\hbox{SL}_{2}(\mathbb{Z})$. We prove the subconvexity bound: for any $\varepsilon>0$,…

We exhibit a class of Schottky subgroups of $\mathbf{PU}(1,n)$ ($n \geq 2$) which we call well-positioned and show that the Hausdorff dimension of the limit set $\Lambda_\Gamma$ associated with such a subgroup $\Gamma$, with respect to the…

We use recent advances on the discretized sum-product problem to obtain new bounds on the Hausdorff dimension of planar $(\alpha,2\alpha)$-Fursterberg sets. This provides a quantitative improvement to the $2\alpha+\epsilon$ bound of…

In this note we derive an upper bound for the Hausdorff dimension of the stable set of a hyperbolic set $\Lambda$ of a $C^2$ diffeomorphisms on a $n$-dimensional manifold. As a consequence we obtain that $\dim_H W^s(\Lambda)=n$ is…

We prove a quantitative statement of the quantum ergodicity for Hecke--Maass cusp forms on the modular surface. As an application of our result, along a density $1$ subsequence of even Hecke--Maass cusp forms, we obtain a sharp lower bound…

We use Kolmogorov complexity methods to give a lower bound on the effective Hausdorff dimension of the point (x, ax+b), given real numbers a, b, and x. We apply our main theorem to a problem in fractal geometry, giving an improved lower…

We find upper and lower bounds on the number of rational points that are $\psi$-approximations of some $n$-dimensional $p$-adic integer. Lattice point counting techniques are used to find the upper bound result, and a Pigeon-hole principle…

We proved that any complete hypersurface in the Euclidean space $\mathbb{R}^{n+1}$ whose Gauss image is contained in an open hemisphere has to be proper. As applications, we derive a counterpart of Hoffman-Osserman-Schoen's result for…

For each $d \in {1,2,3,7,11}$, let $T_d$ be the nearest-integer complex continued fraction map associated with the Euclidean ring $\mathcal{O}*d$, and let $(a_n)$ be its digit sequence. We prove two metric results for this five-system…

The classical Besicovitch-Federer projection theorem implies that the d-dimensional Hausdorff measure of a set in Euclidean space with non-negligible d-unrectifiable part will strictly decrease under orthogonal projection onto almost every…

We study the family of vertical projections whose fibers are right cosets of horizontal planes in the Heisenberg group, $\mathbb{H}^n$. We prove lower bounds for Hausdorff dimension distortion of sets under these mappings, with respect to…

We prove rigidity results involving the Hawking mass for surfaces immersed in a $3$-dimensional, complete Riemannian manifold $(M,g)$ with non-negative scalar curvature (resp. with scalar curvature bounded below by $-6$). Roughly, the main…