Related papers: $\Omega$-results for the hyperbolic lattice point …

Let x be a point in R^2 with irrational slope and let \Gamma denote the lattice SL(2,Z) acting linearly on R^2. Then, the orbit \Gamma x is dense in R^2. We give efective results on the approximation of a point y in R^2 by points of the…

In this paper we derive an explicit formula for the Hirzebruch-Mumford volume of an indefinite lattice L of rank at least 3. If \Gamma is an arithmetic subgroup of the group O(L) of isometries of L and L has signature (2,n), then an…

We evaluate the variance of the number of lattice points in a small randomly rotated spherical ball on a surface of 3-dimensional sphere centered at the origin. Previously, Bourgain, Rudnick, and Sarnak showed conditionally on the…

We consider a class of Hamiltonians in $L^2(\R^2)$ with attractive interaction supported by piecewise $C^2$ smooth loops $\Gamma$ of a fixed length $L$, formally given by $-\Delta-\alpha\delta(x-\Gamma)$ with $\alpha>0$. It is shown that…

We apply the resonance method to obtain large values of general exponential sums with positive coefficients. As applications, we show improved $\Omega$-bounds for Dirichlet and Piltz divisor problems, Gauss circle Problem, and error term…

Let $f:[0,+\infty) \to \mathbb{R}$ be a (locally) Lipschitz function and $\Omega \subset \mathbb{R}^2$ a $C^{1,\alpha}$ domain whose boundary is unbounded and connected. If there exists a positive bounded solution to the overdetermined…

We consider the problem of minimizing the Lagrangian $\int [F(\nabla u)+f\,u]$ among functions on $\Omega\subset\mathbb{R}^N$ with given boundary datum $\varphi$. We prove Lipschitz regularity up to the boundary for solutions of this…

We present examples of geometrically finite manifolds with pinched negative curvature, whose geodesic flow has infinite non-ergodic Bowen-Margulis measure and whose Poincar\'e series converges at the critical exponent $\delta_\Gamma$. We…

Suppose $\Omega\subseteq\RR^d$ is a bounded and measurable set and $\Lambda \subseteq \RR^d$ is a lattice. Suppose also that $\Omega$ tiles multiply, at level $k$, when translated at the locations $\Lambda$. This means that the…

We develop the orbit method in a quantitative form, along the lines of microlocal analysis, and apply it to the analytic theory of automorphic forms. Our main global application is an asymptotic formula for averages of Gan--Gross--Prasad…

A classical result in the study of Ginzburg-Landau equations is that, for Dirichlet or Neumann boundary conditions, if a sequence of functions has energy uniformly bounded on a logarithmic scale then we can find a subsequence whose…

We apply some of the ideas of the Ph.D. Thesis of G. A. Margulis to Teichmuller space. Let x be a point in Teichmuller space, and let B_R(x) be the ball of radius R centered at x (with distances measured in the Teichmuller metric). We…

Heinz Huber (1956) considered the following problem on the the hyperbolic plane H. Consider a strictly hyperbolic subgroup of automorphisms on H with compact quotient, and choose a conjugacy class in this group. Count the number of vertices…

A classical discovery known as Fenchel's conjecture and proved in the 1950s, shows that every co-compact Fuchsian group $F$ has a normal subgroup of finite index isomorphic to the fundamental group of a compact unbordered orientable…

We investigate the Cauchy problem for a class of nonlinear elliptic operators with $C^\infty$-coefficients at a regular set $\Omega \subset R^n$. The Cauchy data are given at a manifold $\Gamma \subset \partial\Omega$ and our goal is to…

We establish an error estimate for counting lattice points in Euclidean norm balls (associated to an arbitrary irreducible linear representation) for lattices in simple Lie groups of real rank at least two. Our approach utilizes refined…

We prove an existence result for Robin boundary value problems modeled on \[ \begin{cases} \Delta u + |\nabla u|^2 + \lambda f(x) = 0 & \text{in } \Omega \\ \frac{\partial u}{\partial \nu} + \beta u = 0 & \text{on } \partial\Omega…

Pointwise error analysis of the linear finite element approximation for $-\Delta u + u = f$ in $\Omega$, $\partial_n u = \tau$ on $\partial\Omega$, where $\Omega$ is a bounded smooth domain in $\mathbb R^N$, is presented. We establish…

Let $\Gamma$ be a group and $r_n(\Gamma)$ the number of its $n$-dimensional irreducible complex representations. We define and study the associated representation zeta function $\calz_\Gamma(s) = \suml^\infty_{n=1} r_n(\Gamma)n^{-s}$. When…

In this note, we study a lattice point counting problem for spheres in Heisenberg groups, incorporating both the non-isotropic dilation structure and the non-commutative group law. More specifically, we establish an upper bound for the…