arXiv subjects

Peter Sarnak

Publications and source records attributed to Peter Sarnak.

At least 37 records · Page 2Linked to original sources

arXiv2015

Local statistics of lattice points on the sphere

A celebrated result of Legendre and Gauss determines which integers can be represented as a sum of three squares, and for those it is typically the case that there are many ways of doing so. These different representations give collections of points on the unit sphere, and a fundamental result, conjectured by Linnik, is that under a simple condition these become uniformly distributed on the sphere. In this note we survey some of our recent work, which explores what happens beyond uniform distribution, giving evidence to randomness on smaller scales. We treat the electrostatic energy, local statistics such as the point pair statistic (Ripley's function), nearest neighbour statistics, minimum spacing and covering radius. We briefly discuss the situation in other dimensions, which is very different. In an appendix we compute the corresponding quantities for random points

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arXiv2014

On the topology of the zero sets of monochromatic random waves

This note concerns the topology of the connected components of the zero sets of monochromatic random waves on compact Riemannian manifolds without boundary. In [SW] it is shown that these are distributed according to a universal measure on the space of smooth topological types. Our goal is to determine the support of this measure.

math-ph↗

arXiv2014

Topologies of nodal sets of random band limited functions

It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In particular the results apply to random monochromatic waves and to random real algebraic hyper-surfaces in projective space.

math-ph↗

arXiv2014

The Möbius function and distal flows

We prove that the Möbius function is linearly disjoint from an analytic skew product on the $2$-torus. These flows are distal and can be irregular in the sense that their ergodic averages need not exist for all points. The previous cases for which such disjointness has been proved are all regular. We also establish the linear disjointness of Möbius from various distal homogeneous flows.

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arXiv2014

The Mobius disjointness conjecture for distal flows

We summarize main results in our paper "The Mobius function and distal flows", and give a direct proof with rate of that the Mobius function is disjoint from Furstenberg's irregular system. This will be published in the Proceedings of the Sixth ICCM, held in Taipei in 2013.

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arXiv2013

Hyperbolic monodromy groups for the hypergeometric equation and Cartan involutions

We give a criterion which ensures that a group generated by Cartan involutions in the automorph group of a rational quadratic form of signature (n-1,1) is "thin", namely it is of infinite index in the latter. It is based on a graph defined on the integral Cartan root vectors, as well as Vinberg's theory of hyperbolic reflection groups. The criterion is shown to be robust for showing that many hyperbolic hypergeometric groups for n_F_(n-1) are thin.

math.GR↗

arXiv2013

Nodal domains of Maass forms I

This paper deals with some questions that have received a lot of attention since they were raised by Hejhal and Rackner in their 1992 numerical computations of Maass forms. We establish sharp upper and lower bounds for the $L^2$-restrictions of these forms to certain curves on the modular surface. These results, together with the Lindelof Hypothesis and known subconvex $L^\infty$-bounds are applied to prove that locally the number of nodal domains of such a form goes to infinity with its eigenvalue.

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arXiv2012

Notes on Thin Matrix Groups

These notes were prepared for the MSRI hot topics workshop on superstrong approximation (2012). We give a brief overview of the developments in the theory, especially the fundamental expansion theorem. Applications to diophantine problems on orbits of integer matrix groups, the affine sieve, group theory, gonality of curves and Heegaard genus of hyperbolic three manifolds, are given. We also discuss the ubiquity of thin matrix groups in various contexts, and in particular that of monodromy groups.

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arXiv2011

Distjointness of Mobius from horocycle flows

We formulate and prove a finite version of Vinogradov's bilinear sum inequality. We use it together with Ratner's joinings theorems to prove that the Mobius function is disjoint from discrete horocycle flows on $Γ\backslash SL_2(\mathbb{R}).

math.NT↗

arXiv2011

Affine Sieve

We establish the main saturation conjecture in [BGS10] connected with executing a Brun sieve in the setting of an orbit of a group of affine linear transformations. This is carried out under the condition that the Zariski closure of the group is Levi-semisimple. It is likely that this condition is also necessary for such saturation to hold.

math.NT↗

arXiv2011

Real zeros of holomorphic Hecke cusp forms

This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments and we call those zeros that lie on these segments, real. Our main results give estimates for the number of real zeros as the weight goes to infinity.

math.NT↗

arXiv2010

Sector estimates for hyperbolic isometries

We prove various orbital counting statements for Fuchsian groups of the second kind. These are of independent interest, and also are used in the companion paper [BourgainKontorovich2009] to produce primes in the Affine Linear Sieve.

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arXiv2009

Generalization of Selberg's 3/16 Theorem and Affine Sieve

A celebrated theorem of Selberg states that for congruence subgroups of SL(2,Z) there are no exceptional eigenvalues below 3/16. We prove a generalization of Selberg's theorem for infinite index "congruence" subgroups of SL(2,Z). Consequently we obtain sharp upper bounds in the affine linear sieve, where in contrast to \cite{BGS} we use an archimedean norm to order the elements.

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arXiv2009

The variance of arithmetic measures associated to closed geodesics on the modular surface

We determine the variance for the fluctuations of the arithmetic measures obtained by collecting all closed geodesics on the modular surface with the same discriminant and ordering them by the latter. This arithmetic variance differs by subtle factors from the variance that one gets when considering individual closed geodesics when ordered by their length. The arithmetic variance is the same one that appears in the fluctuations of measures associated with quantum states on the modular surface.

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arXiv2009

Strong Spectral Gaps for Compact Quotients of Products of $\PSL(2,\bbR)$

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.

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arXiv2009

Prime and almost prime integral points on principal homogeneous spaces

We develop the affine sieve in the context of orbits of congruence subgroups of semi-simple groups acting linearly on affine space. In particular we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors. In many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable and the orbit is the integers. When the orbit is the set of integral matrices of a fixed determinant we obtain a sharp result for the saturation number, and thus establish the Zariski density of matrices all of whose entries are prime numbers. Among the key tools used are explicit approximations to the generalized Ramanujan conjectures for such groups and sharp and uniform counting of points on such orbits when ordered by various norms.

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arXiv2003

Arithmetic and equidistribution of measures on the sphere

Motivated by problems of mathematical physics (quantum chaos) questions of equidistribution of eigenfunctions of the Laplace operator on a Riemannian manifold have been studied by several authors. We consider here, in analogy with arithmetic hyperbolic surfaces, orthonormal bases of eigenfunctions of the Laplace operator on the two dimensional unit sphere which are also eigenfunctions of an algebra of Hecke operators which act on these spherical harmonics. We formulate an analogue of the equidistribution of mass conjecture for these eigenfunctions as well as of the conjecture that their moments tend to moments of the Gaussian as the eigenvalue increases. For such orthonormal bases we show that these conjectures are related to the analytic properties of degree eight arithmetic L-functions associated to triples of eigenfunctions. Moreover we establish the conjecture for the third moments and give a conditional (on standard analytic conjectures about these arithmetic L-functions) proof of the equdistribution of mass conjecture.

math.NT↗

arXiv2001

The distribution of spacings between the fractional parts of $n^2 α$

We study the distribution of normalized spacings between the fractional parts of an^2, n=1,2,.... We conjecture that if a is "badly approximable" by rationals, then the sequence of fractional parts has Poisson spacings, and give a number of results towards this conjecture. We also present an example of a Diophantine number a for which the higher correlation functions of the sequence blow up.

math.NT↗