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Alex Kontorovich

Publications and source records attributed to Alex Kontorovich.

At least 19 recordsLinked to original sources
arXiv2026

Arithmetic Polyhedra

The Koebe-Andreev-Thurston theorem assigns a 3-dimensional hyperbolic reflection group to each combinatorial polyhedron. A natural question is: which of them are arithmetic? In 2016, Kontorovich-Nakamura conjectured that all arithmetic reflection groups obtained in this way are commensurable to those obtained from the tetrahedron, square pyramid, or cuboctahedron. In this paper, we prove the conjecture. It is a consequence of the following result of independent interest: all arithmetic ideal, right-angled hyperbolic polyhedra are obtained by gluing together copies of one of three ``seed'' polyhedra.

math.MG↗
arXiv2026

Lax-Phillips orbit counting in higher rank

Given a discrete lattice, $Γ< \operatorname{SL}_m(\mathbb{R})$, and a base point $o \in \mathbb{R}^m$, let $N_Γ(T)$ denote the number of points in the orbit $o \cdot Γ$ whose (Euclidean) length is bounded by a growing parameter, $T$. We demonstrate an abstract spectral method à la Lax-Phillips, capable of obtaining strong asymptotic estimates for $N_Γ(T)$, and compare and contrast it with other methods.

math.NT↗
arXiv2026

Diamond Open Access: The AMR Experiment

Diamond open access journals charge neither readers nor authors. Despite long-standing support for this ideal within mathematics, relatively few such journals exist. This article documents the Association for Mathematical Research's experience building and operating diamond open access journals, focusing on the infrastructure, cost, and editorial practices that make the model viable. It aims to clarify why earlier reform efforts have been difficult to replicate and how a lightweight institutional framework can lower the barrier to adoption.

math.HO↗
arXiv2025

The Shape of Math To Come

We present an overview of how certain computational tools currently interact with mathematical practice, and reflect on the implications for research mathematics in the short to medium term, as the field navigates the emerging age of AI and formal verification systems.

math.HO↗
arXiv2025

Spanning trees and continued fractions

We prove the exponential growth of the cardinality of the set of numbers of spanning trees in simple (and planar) graphs on $n$ vertices, answering a question of Sedláček from 1969. The proof uses a connection with continued fractions, ``thin orbits,'' and Zaremba's conjecture.

math.CO↗
arXiv2025

Effective resistance in planar graphs and continued fractions

For a simple graph $G=(V,E)$ and edge $e\in E$, the effective resistance is defined as a ratio $\frac{τ(G/e)}{τ(G)}$, where $τ(G)$ denotes the number of spanning trees in $G$. We resolve the inverse problem for the effective resistance for planar graphs. Namely, we determine (up to a constant) the smallest size of a simple planar graph with a given effective resistance. The results are motivated and closely related to our previous work arXiv:2411.18782 on Sedláček's inverse problem for the number of spanning trees.

math.CO↗
arXiv2025

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.

math.NT↗
arXiv2025

Sign changes along geodesics of modular forms

Given a compact segment, $β$, of a cuspidal geodesic on the modular surface, we study the number of sign changes of cusp forms and Eisenstein series along $β$. 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 Lindelöf hypothesis from recent work of Ki [Ki23].

math.NT↗
arXiv2024

Kleinian sphere packings, reflection groups, and arithmeticity

In this paper we study crystallographic sphere packings and Kleinian sphere packings, introduced first by Kontorovich and Nakamura in 2017 and then studied further by Kapovich and Kontorovich in 2021. In particular, we solve the problem of existence of crystallographic sphere packings in certain higher dimensions posed by Kontorovich and Nakamura. In addition, we present a geometric doubling procedure allowing to obtain sphere packings from some Coxeter polyhedra without isolated roots, and study "properly integral" packings (that is, ones which are integral but not superintegral). Our techniques rely extensively on computations with Lorentzian quadratic forms, their orthogonal groups, and associated higher-dimensional hyperbolic polyhedra.

math.GT↗
arXiv2023

Notes on a Path to AI Assistance in Mathematical Reasoning

These informal notes are based on the author's lecture at the National Academies of Science, Engineering, and Mathematics workshop on "AI to Assist Mathematical Reasoning" in June 2023. The goal is to think through a path by which we might arrive at AI that is useful for the research mathematician.

math.HO↗
arXiv2022

Effective counting in sphere packings

Given a Zariski-dense, discrete group, $Γ$, of isometries acting on $(n + 1)$-dimensional hyperbolic space, we use spectral methods to obtain a sharp asymptotic formula for the growth rate of certain $Γ$-orbits. In particular, this allows us to obtain a best-known effective error rate for the Apollonian and (more generally) Kleinian sphere packing counting problems, that is, counting the number of spheres in such with radius bounded by a growing parameter. Our method extends the method of Kontorovich [Kon09], which was itself an extension of the orbit counting method of Lax-Phillips [LP82], in two ways. First, we remove a compactness condition on the discrete subgroups considered via a technical cut-off and smoothing operation. Second, we develop a coordinate system which naturally corresponds to the inversive geometry underlying the sphere counting problem, and give structure theorems on the arising Casimir operator and Haar measure in these coordinates.

math.GT↗
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.

math.SP↗
arXiv2022

On Length Sets of Subarithmetic Hyperbolic Manifolds

We formulate the Asymptotic Length-Saturation Conjecture on the length sets of closed geodesics on hyperbolic manifolds whose fundamental groups are subarithmetic, that is, contained in an arithmetic group. We prove the first instance of the conjecture for punctured, Zariski dense covers of the modular surface. The tools involved include the Orbital Circle Method, expansion and counting in congruence towers of thin groups, estimates for exponential sums, bilinear forms, and quadratic L-series.

math.NT↗
arXiv2021

On Superintegral Kleinian Sphere Packings, Bugs, and Arithmetic Groups

We develop the notion of a Kleinian Sphere Packing, a generalization of "crystallographic" (Apollonian-like) sphere packings defined by Kontorovich-Nakamura [KN19]. Unlike crystallographic packings, Kleinian packings exist in all dimensions, as do "superintegral" such. We extend the Arithmeticity Theorem to Kleinian packings, that is, the superintegral ones come from Q-arithmetic lattices of simplest type. The same holds for more general objects we call Kleinian Bugs, in which the spheres need not be disjoint but can meet with dihedral angles pi/m for finitely many m. We settle two questions from [KN19]: (i) that the Arithmeticity Theorem is in general false over number fields, and (ii) that integral packings only arise from non-uniform lattices.

math.NT↗
arXiv2018

What Is... A Thin Group?

This paper describes in basic terms what a "Thin Group" is, as well as its uses in various subjects.

math.NT↗
arXiv2018

On Toric Orbits in the Affine Sieve

We give a detailed analysis of a heuristic model for the failure of "saturation" in instances of the Affine Sieve having toral Zariski closure. Based on this model, we formulate precise conjectures on several classical problems of arithmetic interest, and test these against empirical data.

math.NT↗