Related papers: The lattice point counting problem on the Heisenbe…
We obtain the $n$th centered moments of one level densities of a large orthogonal family of $L$-functions associated with holomorphic Hecke newforms of level $q$, averaged over $q\sim Q$. We verify the Katz-Sarnak conjecture for these…
Refining an earlier result due to Hahlomaa, we provide a new Carleson-type condition for $k$-regular sets in the Heisenberg group $\mathbb{H}^n$ to have big pieces of Lipschitz images of subsets of $\mathbb{R}^k$ for $1\leq k\leq n$. Our…
We develop regularity theory for critical points of variational integrals defined on Hessian spaces of functions on open, bounded subdomains of $\mathbb{R}^n$, under compactly supported variations. The critical point solves a fourth order…
The Gauss circle problem asks for an approximation to the number of lattice points of $\mathbb{Z}^2$ contained in $B_r$, the disk of radius $r$ centered at the origin. Upper, lower, and average bounds have been established for this…
We prove a sharp bound for the remainder term of the number of lattice points inside a ball, when averaging over a compact set of (not necessarily unimodular) lattices, in dimensions two and three. We also prove that such a bound cannot…
The problem of computing the index of a coincidence isometry of the hyper cubic lattice $\mathbb{Z}^{n}$ is considered. The normal form of a rational orthogonal matrix is analyzed in detail, and explicit formulas for the index of certain…
The left-invariant sub-Lorentzian problem on the Heisenberg group is considered. An optimal synthesis is constructed, the sub-Lorentzian distance and spheres are described.
In this paper we aim at identifying the level sets of the gauge norm in the Heisenberg group $\mathbb{H}^n$ via the prescription of their (non-constant) horizontal mean curvature. We establish a uniqueness result in $\mathbb{H}^1$ under an…
The Heisenberg scaling is typically associated with nonclassicality and entanglement. In this work, however, we discuss how classical long-range correlations between lattice sites in many-body systems may lead to a 1/N scaling in precision…
We consider the three-dimensional Heisenberg group, equipped with any left-invariant metric, either Lorentzian or Riemannian. We completely classify their affine vector fields and investigate their relationship with Killing vector fields…
We compute the first and second moment formulas for Siegel transforms related to problems counting primitive lattice points in the real plane with congruence conditions. As applications, we derive an analog of Schmidt's random counting…
We study the asymptotic distribution of norm ball averages along orbits of a lattice $\Gamma \subset \text{SO}(n,1)$ acting on the moduli space of pairs of orthogonal discrete subgroups of $\mathbb{R}^{n+1}$ up to homothety. Our main result…
We generalise M. M. Skriganov's notion of weak admissibility for lattices to include standard lattices occurring in Diophantine approximation and algebraic number theory, and we prove estimates for the number of lattice points in sets such…
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 introduce maximal and average coherence on lattices by analogy with these notions on frames in Euclidean spaces. Lattices with low coherence can be of interest in signal processing, whereas lattices with high orthogonality defect are of…
We prove a quantitative estimate with a power saving error term for the number of points in a mapping class group orbit of Teichm\"uller space that lie within a Teichm\"uller metric ball of given center and large radius. Estimates of the…
In this paper we study various Hardy inequalities in the Heisenberg group $\mathbb H^n$, w.r.t. the Carnot-Carath\'eodory distance $\delta$ from the origin. We firstly show that the optimal constant for the Hardy inequality is strictly…
For a convex body B in three-dimensional Euclidean space, which is invariant under rotations around one coordinate axis and has a smooth boundary of bounded nonzero curvature, the lattice point discrepancy (number of integer points minus…
We study the fine distribution of lattice points lying on expanding circles in the hyperbolic plane $\mathbb{H}$. The angles of lattice points arising from the orbit of the modular group $PSL_{2}(\mathbb{Z})$, and lying on hyperbolic…
We consider the lattice point problem corresponding to a family of elliptic paraboloids in $\mathbb{R}^d$ with $d\ge3$ and we prove the expected to be optimal exponent, improving previous results. This is especially noticeable for $d=3$…