What Is... A Thin Group?
This paper describes in basic terms what a "Thin Group" is, as well as its uses in various subjects.
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arXiv subjects
Alex Kontorovich
Publications and source records attributed to Alex Kontorovich.
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.
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Beyond Expansion IV: Traces of Thin Semigroups
We continue our study of particular instances of the Affine Sieve, producing levels of distribution beyond those attainable from expansion alone. Motivated by McMullen's Arithmetic Chaos Conjecture regarding low-lying closed geodesics on the modular surface defined over a given number field, we study the set of traces for certain sub-semi-groups of SL2(Z) corresponding to absolutely Diophantine numbers. In particular, we are concerned with the level of distribution for this set. While the standard Affine Sieve procedure, combined with Bourgain-Gamburd-Sarnak's resonance-free region for the resolvent of a "congruence" transfer operator, produces some exponent of distribution alpha > 0, we are able to produce the exponent alpha < 1/3. This recovers unconditionally the same exponent as what one would obtain under a Ramanujan-type conjecture for thin groups. A key ingredient, of independent interest, is a bound on the additive energy of SL2(Z).
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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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Geometry and Arithmetic of Crystallographic Sphere Packings
We introduce the notion of a "crystallographic sphere packing," defined to be one whose limit set is that of a geometrically finite hyperbolic reflection group in one higher dimension. We exhibit for the first time an infinite family of conformally-inequivalent such with all radii being reciprocals of integers. We then prove a result in the opposite direction: the "superintegral" ones exist only in finitely many "commensurability classes," all in dimensions below 30.
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The Local-Global Principle for Integral Soddy Sphere Packings
Fix an integral Soddy sphere packing P. Let K be the set of all curvatures in P. A number n is called represented if n is in K, that is, if there is a sphere in P with curvature equal to n. A number n is called admissible if it is everywhere locally represented, meaning that n is in K(mod q) for all q. It is shown that every sufficiently large admissible number is represented.
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Efficiently constructing tangent circles
Our goal is to present, in what we believe is the most efficient way possible, a construction of four mutually tangent circles.
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Beyond Expansion III: Reciprocal Geodesics
We prove the existence of infinitely many low-lying and fundamental closed geodesics on the modular surface which are reciprocal, that is, invariant under time reversal. The method combines ideas from Parts I and II of this series, namely the dispersion method in bilinear forms, as applied to thin semigroups coming from restricted continued fractions.
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Beyond Expansion II: Low-Lying Fundamental Geodesics
A closed geodesic on the modular surface is "low-lying" if it does not travel "high" into the cusp. It is "fundamental" if it corresponds to an element in the class group of a real quadratic field. We prove the existence of infinitely many low-lying fundamental geodesics, answering a question of Einsiedler-Lindenstrauss-Michel-Venkatesh.
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Applications of Thin Orbits
This text is based on a series of three expository lectures on a variety of topics related to "thin orbits," as delivered at Durham University's Easter School on "Dynamics and Analytic Number Theory" in April 2014. The first lecture reviews closed geodesics on the modular surface and the reduction theory of binary quadratic forms before discussing Duke's equidistribution theorem (for indefinite classes). The second lecture exposits three quite different but (it turns out) not unrelated problems, due to Einsiedler-Lindenstrauss-Michel-Venkatesh, McMullen, and Zaremba. The third lecture reformulates these in terms of the aforementioned thin orbits, and shows how all three would follow from a single "Local-Global" Conjecture of Bourgain and the author. We also describe some partial progress on the conjecture, which has lead to some results on the original problems.
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Thermodynamic expansion to arbitrary moduli
We extend the thermodynamic expansion results in [BourgainGamburdSarnak2011, MageeOhWinter2015] from square-free to arbitrary moduli by developing a novel decoupling technique and applying [BourgainVarju2012].
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Effective equidistribution of shears and applications
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.
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Levels of Distribution and the Affine Sieve
This article is an expanded version of the author's lecture in the Basic Notions Seminar at Harvard, September 2013. Our goal is a brief and introductory exposition of aspects of two topics in sieve theory which have received attention recently: (1) the spectacular work of Yitang Zhang, under the title "Level of Distribution," and (2) the so-called "Affine Sieve," introduced by Bourgain-Gamburd-Sarnak.
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On the Pair Correlation Density for Hyperbolic Angles
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.
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Almost Prime Coordinates for Anisotropic and Thin Pythagorean Orbits
We make an observation which doubles the exponent of distribution in certain Affine Sieve problems, such as those considered by Liu-Sarnak, Kontorovich, and Kontorovich-Oh. As a consequence, we decrease the known bounds on the saturation numbers in these problems.
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On Zaremba's Conjecture
It is shown that there is a constant A and a density one subset S of the positive integers, such that for all q in S there is some 1<=p<q, (p, q)=1, so that p/q has all its partial quotients bounded by A.
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On Zaremba's Conjecture
Zaremba's 1971 conjecture predicts that every integer appears as the denominator of a finite continued fraction whose partial quotients are bounded by an absolute constant. We confirm this conjecture for a set of density one.
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The Affine Sieve Beyond Expansion I: Thin Hypotenuses
We study an instance of the Affine Sieve, producing a level of distribution beyond that which can be obtained from current techniques, even assuming a Selberg/Ramanujan-type spectral gap. In particular, we consider the set of hypotenuses in a thin orbit of Pythagorean triples. Previous work [Kon07, Kon09, KO12] gave an exponent of distribution alpha < 1/12 coming from Gamburd's [Gam02] gap theta = 5/6, thereby producing R = 13 almost primes in this linear sieve problem (see Sec. 1 for definitions). If conditioned on a best possible gap theta = 1/2, the known method would give an exponent alpha < 1/4, and R = 5 almost primes. The exponent 1/4 is the natural analogue of the "Bombieri-Vinogradov" range of distribution for this problem, see Remark 1.19. In this paper, we unconditionally prove the exponent alpha < 7/24 (in the "Elliott-Halberstam" range), thereby producing R = 4 almost primes. The main tools involve developing bilinear forms and the dispersion method in the range of incomplete sums for this Affine Sieve problem.
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