arXiv subjects
Publications and source records attributed to Dubi Kelmer.
We give upper and lower bounds for Diophantine exponents measuring how well a point in the plane can be approximated by points in the orbit of a lattice $Γ<\mathrm{SL}_2(\mathbb{R})$ acting linearly on $\mathbb{R}^2$. Our method gives bounds that are uniform for almost all orbits.
We prove a quantitative version of Oppenheim's conjecture for generic ternary indefinite quadratic forms. Our results are inspired by and analogous to recent results for diagonal quadratic forms due to Bourgain.
We study the shrinking target problem for actions of semisimple groups on homogeneous spaces, with applications to logarithm laws and Diophantine approximation.
We give mean square bounds for the remainder in the lattice point counting problem, counting the number of lattice points in a large ball in $\mathbb{R}^d$, when averaged over families of shears of the lattice.
Given a unimodular lattice $Λ\subseteq \mathbb{R}^2$ consider the counting function $\mathcal{N}_Λ(T)$ counting the number of lattice points of norm less than $T$, and the remainder $\mathcal{R}_Λ(T)=\mathcal{N}(T)-πT^2$. We give an elementary proof that the mean square of the remainder over the set of all shears of a unimodular lattice is bounded by $O(T\log^2(T))$.
A "shear" is a unipotent translate of a cuspidal geodesic ray in the quotient of the hyperbolic plane by a non-uniform discrete subgroup of PSL(2,R), possibly of infinite co-volume. We prove the regularized equidistribution of shears under large translates with effective (that is, power saving) rates. We also give applications to weighted second moments of GL(2) automorphic L-functions, and to counting lattice points on locally affine symmetric spaces.
We extend results of Bhagwat and Rajan on a strong multiplicity one property for length spectrum to hyperbolic manifolds with cusps, showing that for two even dimensional hyperbolic manifolds of finite volume, if all but finitely many closed geodesics have the same length, then all closed geodesics have the same length. When the set of exceptional lengths is infinite, but sufficiently sparse, we can show that the two manifolds must have the same volume, and in low dimensions also the same number of cusps. A main ingredient in our proof is a generalization of a result of Selberg on the distribution of poles of Eisenstein series to hyperbolic manifolds.
Let $Γ< \mathrm{PSL}_2(\mathbb{R})$ be a lattice and $ω\in \mathbb{H}$ a point in the upper half plane. We prove the existence and give an explicit formula for the pair correlation density function for the set of angles between geodesic rays of the lattice $Γω$ intersected with increasingly large balls centered at $ω$, thus proving a conjecture of Boca-Popa-Zaharescu.
In a series of lectures Selberg introduced a trace formula on the space of hybrid Maass-modular forms of an irreducible uniform lattice in $\PSL_2(\bbR)^n$. In this paper we derive the analogous formula for a non-uniform lattice and use it to study the distribution of elliptic-hyperbolic conjugacy classes. In particular, for Hilbert modular groups there is a nice interpretation of these results in terms of class numbers and fundamental units of quadratic forms over totally real number fields.
A closed geodesic on the modular surface gives rise to a knot on the 3-sphere with a trefoil knot removed, and one can compute the linking number of such a knot with the trefoil knot. We show that, when ordered by their length, the set of closed geodesics having a prescribed linking number become equidistributed on average with respect to the Liouville measure. We show this by using the thermodynamic formalism to prove an equidistribution result for a corresponding set of quadratic irrationals on the unit interval.
Let $\calM_1$ and $\calM_2$ denote two compact hyperbolic manifolds. Assume that the multiplicities of eigenvalues of the Laplacian acting on $L^2(\calM_1)$ and $L^2(\calM_2)$ (respectively, multiplicities of lengths of closed geodesics in $\calM_1$ and $\calM_2$) are the same, except for a possibly infinite exceptional set of eigenvalues (respectively lengths). We define a notion of density for the exceptional set and show that if it is below a certain threshold, the two manifolds must be iso-spectral.
Let $G$ be $\SO(n,1)$ or $\SU(n,1)$ and let $Γ\subset G$ denote an arithmetic lattice. The hyperbolic manifold $Γ\backslash \calH$ comes with a natural family of covers, coming from the congruence subgroups of $Γ$. In many applications, it is useful to have a bound for the spectral gap that is uniform for this family. When $Γ$ is itself a congruence lattice, there are very good bounds coming from known results towards the Ramanujan conjectures. In this paper, we establish an effective bound that is uniform for congruence subgroups of a non-congruence lattice.
Let $\calM=Γ\bs \calH^{(n)}$, where $\calH^{(n)}$ is a product of $n+1$ hyperbolic planes and $Γ\subset\PSL(2,\bbR)^{n+1}$ is an irreducible cocompact lattice. We consider closed geodesics on $\calM$ that propagate locally only in one factor. We show that, as the length tends to infinity, the holonomy rotations attached to these geodesics become equidistributed in $\PSO(2)^n$ with respect to a certain measure. For the special case of lattices derived from quaternion algebras, we can give another interpretation of the holonomy angles under which this measure arises naturally.
The existence of a strong spectral gap for lattices in semi-simple Lie groups is crucial in many applications. In particular, for arithmetic lattices it is useful to have bounds for the strong spectral gap that are uniform in the family of congruence covers. When the lattice is itself a congruence group, there are uniform and very good bounds for the spectral gap coming from the known bounds toward the Ramanujan-Selberg conjectures. In this note, we establish a uniform bound for the strong spectral gap for congruence covers of an irreducible co-compact lattice $Γ$ in $PSL(2,R)^d$ with $d\geq 2$, which is the simplest and most basic case where the congruence subgroup property is not known.
The existence of a strong spectral gap for quotients $Γ\bs G$ of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the Ramanujan-Selberg Conjectures. If $G$ has no compact factors then for general lattices a strong spectral gap can still be established, however, there is no uniformity and no effective bounds are known. This note is concerned with the strong spectral gap for an irreducible co-compact lattice $Γ$ in $G=\PSL(2,\bbR)^d$ for $d\geq 2$ which is the simplest and most basic case where the congruence subgroup property is not known. The method used here gives effective bounds for the spectral gap in this setting.
The quantum cat map is a model for a quantum system with underlying chaotic dynamics. In this paper we study the matrix elements of smooth observables in this model, when taking arithmetic symmetries into account. We give explicit formulas for the matrix elements as certain exponential sums. With these formulas we can show that there are sequences of eigenfunctions for which the matrix elements decay significantly slower then was previously conjectured. We also prove a limiting distribution for the fluctuation of the normalized matrix elements around their average.
For manifolds with geodesic flow that is ergodic on the unit tangent bundle, the quantum ergodicity theorem implies that almost all Laplacian eigenfunctions become equidistributed as the eigenvalue goes to infinity. For a locally symmetric space with a universal cover that is a product of several upper half planes, the geodesic flow has constants of motion so it can not be ergodic. It is, however, ergodic when restricted to the submanifolds defined by these constants. In accordance, we show that almost all eigenfunctions become equidistributed on these submanifolds.
In this note we study Kloosterman sums twisted by a multiplicative characters modulo a prime power. We show, by an elementary calculation, that these sums become equidistributed on the real line with respect to a suitable measure.