arXiv subjects

Dubi Kelmer

Publications and source records attributed to Dubi Kelmer.

At least 19 recordsLinked to original sources

arXiv2026

Strong spectral gap for geometrically finite hyperbolic manifolds

Let $\Gamma < G := \operatorname{SO}(d+1, 1)$ for $d \geq 1$ be a Zariski dense, geometrically finite, discrete subgroup with critical exponent strictly greater than $d/2$. We show that $L^2(\Gamma\backslash G)$ admits a strong spectral gap, confirming a conjecture of Mohammadi and Oh. This extends the spherical spectral gap on $L^2(\Gamma\backslash \mathbb{H}^{d+1}) \cong L^2(\Gamma\backslash G/\operatorname{SO}(d+1))$, which follows by the works of Lax-Phillips, Patterson, and Sullivan by different methods. As a consequence, we establish rates of decay of matrix coefficients, and of exponential mixing of the frame flow, that are explicitly determined by the size of the strong spectral gap.

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arXiv2025

The density Hypothesis for irreducible lattices in $\mathrm{PSL}_2(\mathbb{R})^d$

We prove the density hypothesis for congruence subgroups of an irreducible uniform lattice in $\mathrm{PSL}_2(\mathbb{R})^d$, extending previous results on the spherical density hypothesis to bound multiplicities of non-tempered non-spherical representations. Our bounds are uniform in the level as well as the spectral parameters.

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arXiv2024

Sign changes along geodesics of modular forms

Given a compact segment, $\beta$, of a cuspidal geodesic on the modular surface, we study the number of sign changes of cusp forms and Eisenstein series along $\beta$. We prove unconditionally a sharp lower bound for Eisenstein series along a full density set of spectral parameters. Conditioned on certain moment bounds, we extend this to all spectral parameters, and prove similar theorems for cusp forms. The arguments rely in part on the authors' mean square bounds [KKL24], and on removing the assumption of the Lindel\"of hypothesis from recent work of Ki [Ki23].

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arXiv2024

Effective density of values of indefinite ternary inhomogeneous quadratic forms

Given an inhomogeneous quadratic form $Q_ξ(v)=Q(v+ξ)$ with $Q$ an indefinite $\mathbb{Q}$-isotropic rational ternary form and $ξ\in \mathbb{R}^3$ irrational, we prove an effective lower bound for the number of integer vectors $v\in \mathbb{Z}^n$ with $\|v\| \leq T$ such that $|Q_ξ(v)-t|<δ$ that is valid for any $t\in \mathbb{R}$ and all $δ\geq T^{-ν}$, with $ν>0$ depending explicitly on the Diophantine properties of $ξ$. In particular, for $ξ$ with algebraic entries we can take any $ν<\frac{1}{8}$.

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arXiv2024

Counting rational points on the sphere with bounded denominator

We give an optimal bound for the remainder when counting the number of rational points on the $n$-dimensional sphere with bounded denominator for any $n\geq 2$.

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arXiv2023

Fourier expansion of light-cone Eisenstein series

In this work we give an explicit formula for the Fourier coefficients of Eisenstein series corresponding to certain arithmetic lattices acting on hyperbolic n+1-space. As a consequence we obtain results on location of all poles of these Eisenstein series as well as their supremum norms. We use this information to get new results on counting rational points on spheres.

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arXiv2023

Second moment of the light-cone Siegel transform and applications

We study the light-cone Siegel transform, transforming functions on the light cone of a rational indefinite quadratic form $Q$ to a function on the homogenous space $\text{SO}^+_Q(\mathbb{Z})\backslash \text{SO}^+_Q(\mathbb{R})$. In particular, we prove a second moment formula for this transform for forms of signature $(n+1,1)$, and show how it can be used for various applications for counting integer points on the light cone. In particular, we prove some new results on intrinsic Diophantine approximations on ellipsoids as well as on the distribution of values of random linear and quadratic forms on the light cone.

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arXiv2023

Norm bounds on Eisenstein series

We study the sup-norm and mean-square-norm problems for Eisenstein series on certain arithmetic hyperbolic orbifolds, producing sharp exponents for the modular surface and Picard 3-fold. The methods involve bounds for Epstein zeta functions, and counting restricted values of indefinite quadratic forms at integer points.

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arXiv2022

Sarnak's spectral gap question

We answer in the affirmative a question of Sarnak's from 2007, confirming that the Patterson-Sullivan base eigenfunction is the unique square-integrable eigenfunction of the hyperbolic Laplacian invariant under the group of symmetries of the Apollonian packing. Thus the latter has a maximal spectral gap. We prove further restrictions on the spectrum of the Laplacian on a wide class of manifolds coming from Kleinian sphere packings.

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arXiv2022

Effective density for inhomogeneous quadratic forms II: fixed forms and generic shifts

We establish effective versions of Oppenheim's conjecture for generic inhomogeneous quadratic forms. We prove such results for fixed quadratic forms and generic shifts. Our results complement our companion paper where we considered generic forms and fixed shifts. In this paper, we use ergodic theorems and in particular we establish a strong spectral gap with effective bounds for some representations of orthogonal groups which do not possess Kazhdan's property (T).

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arXiv2021

Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds

Let $\mathcal{M}$ be a geometrically finite hyperbolic manifold. We present a very general theorem on the shrinking target problem for the geodesic flow, using its exponential mixing. This includes a strengthening of Sullivan's logarithm law for the excursion rate of the geodesic flow. More generally, we prove logarithm laws for the first hitting time for shrinking cusp neighborhoods, shrinking tubular neighborhoods of a closed geodesic, and shrinking metric balls, as well as give quantitative estimates for the time a generic geodesic spends in such shrinking targets.

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arXiv2020

Effective density for inhomogeneous quadratic forms I: generic forms and fixed shifts

We establish effective versions of Oppenheim's conjecture for generic inhomogeneous quadratic forms. We prove such results for fixed shift vectors and generic quadratic forms. When the shift is rational we prove a counting result which implies the optimal density for values of generic inhomogeneous forms. We also obtain a similar density result for fixed irrational shifts satisfying an explicit Diophantine condition. The main technical tool is a formula for the second moment of Siegel transforms on certain congruence quotients of $\operatorname{SL}_n(\mathbb{R})$ which we believe to be of independent interest. In a sequel, we use different techniques to treat the companion problem concerning generic shifts and fixed quadratic forms.

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arXiv2018

Values of random polynomials in shrinking targets

Relying on the classical second moment formula of Rogers we give an effective asymptotic formula for the number of integer vectors $v$ in a ball of radius $t$, with value $Q(v)$ in a shrinking interval of size $t^{-κ}$, that is valid for almost all indefinite quadratic forms in $n$ variables for any $κ<n-2$. This implies in particular, the existence of such integer solutions establishing the prediction made by Ghosh Gorodnik and Nevo. We also obtain similar results for random polynomials of higher degree.

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arXiv2018

Exponents for the Equidistribution of Shears and Applications

In previous work, the authors introduced "soft" methods to prove the effective (i.e. with power savings error) equidistribution of "shears" in cusped hyperbolic surfaces. In this paper, we study the same problem but now allow full use of the spectral theory of automorphic forms to produce explicit exponents, and uniformity in parameters. We give applications to counting square values of quadratic forms.

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arXiv2018

The second moment of the Siegel transform in the space of symplectic lattices

Using results from spectral theory of Eisenstein series, we prove a formula for the second moment of the Siegel transform when averaged over the subspace of symplectic lattices. This generalizes the classical formula of Rogers for the second moment in the full space of unimodular lattices. Using this new formula we give very strong bounds for the discrepancy of the number of lattice points in an Borel set, which hold for generic symplectic lattices.

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arXiv2017

Shrinking targets problems for flows on homogeneous spaces

We study shrinking targets problems for discrete time flows on a homogenous space $Γ\backslash G$ with $G$ a semisimple group and $Γ$ an irreducible lattice. Our results apply to both diagonalizable and unipotent flows, and apply to very general families of shrinking targets. As a special case, we establish logarithm laws for cusp excursions of unipotent flows answering a question of Athreya and Margulis.

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arXiv2017

Shrinking targets for discrete time flows on hyperbolic manifolds

We prove dynamical Borel Canteli Lemmas for discrete time homogenous flows hitting a sequence of shrinking targets in a hyperbolic manifold. These results apply to both diagonalizable and unipotent flows, and any family of measurable shrinking targets. As a special case, we establish logarithm laws for the first hitting times to shrinking balls and shrinking cusp neighborhoods, refining and improving on perviously known results.

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arXiv2016

Logarithm laws for one parameter unipotent flows

We prove logarithm laws and shrinking target properties for unipotent flows on the homogenous space $Γ\bs G$ with $G=\SL_2(\bbR)^{r_1}\times\SL_2(\bbC)^{r_2}$ and $Γ\subseteq G$ an irreducible non-uniform lattice. Our method relies on certain estimates for the norms of (incomplete) theta series in this setting.

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