arXiv subjects

Zeev Rudnick

Publications and source records attributed to Zeev Rudnick.

At least 19 recordsLinked to original sources

arXiv2026

Closed geodesics in homology classes on random hyperbolic surfaces of large genus

We study the distribution of closed geodesics in homology classes on random hyperbolic surfaces of large genus. Viewing the surface as a random point in moduli space equipped with the Weil--Petersson probability measure, we investigate the fluctuations of the weighted counting function of closed geodesics in homology classes modulo $q$. We show that, in the large genus limit, the variance is asymptotic to $X\log X$ for every modulus $q>2$, with an exceptional factor of two when $q=2$. We relate our findings to a conjecture of Hooley, and to work of Friedlander and Goldston, on primes in arithmetic progressions. We also introduce a statistical model based on the random allocation of weighted balls which exhibits analogous behavior.

math.GT↗

arXiv2026

The classification of real quadratic fields which satisfy Hammarhjelm's condition

A real quadratic field satisfies Hammarhjelm's condition if its ring of integers has unique factorization, and the Minkowski lattice of its ring of integers contains no point in a certain rectangle determined by the fundamental unit. Such fields have recently appeared in the study of visible points in algebraic cut-and-project sets. We prove that there are exactly seven real quadratic fields satisfying Hammarhjelm's condition, namely those with discriminant 8, 5, 13, 29, 53, 173, 293. The proof is based on showing that for such fields, the fundamental unit is small relative to the discriminant, together with genus theory and Biro's classification of class number one fields in Yokoi's family.

math.NT↗

arXiv2026

Closed geodesics in short intervals for random hyperbolic surfaces

We study the distribution of closed geodesics in short intervals on random hyperbolic surfaces of large genus, and compare it with the classical problem of primes in short intervals. Viewing the surface $M$ as a random point in moduli space equipped with the Weil--Petersson measure, we investigate the random variable $\Psi_M(x;H)$ counting closed geodesics with norms in the interval $[X, X+H]$, weighted by primitive length, where $H=o(X)$. This is analogous to the Chebyshev function in prime number theory. Our main result establishes that in the large genus limit, \[ \lim_{g\to \infty}\mathrm{Var}(\Psi_M(X;H)) \sim 2\,H \log X, \] when $X\to \infty$, $H=o(X)$. Goldston and Montgomery related the variance for primes in short intervals to the form factor associated with zeros of the Riemann zeta function, and conjectured that it is asymptotic to \[ H\log(X/H). \] We show that for automorphic L-functions of degree $d>1$, the early-time GUE form factor already follows from the Riemann Hypothesis, thereby recovering the variance $H\log X$ in the very short interval regime predicted by Bui, Keating and Smith. In the geometric setting, the appearance of $\log X$ reflects the much higher spectral density of Laplace eigenvalues relative to zeros of finite-degree $L$-functions, while the additional factor of $2$ is explained by the expected GOE statistics for the Laplace spectrum of generic hyperbolic surfaces.

math.GT↗

arXiv2024

On quantum ergodicity for higher dimensional cat maps

We study eigenfunction localization for higher dimensional cat maps, a popular model of quantum chaos. These maps are given by linear symplectic maps in ${\mathrm{Sp}}(2g,\mathbb Z)$, which we take to be ergodic. Under some natural assumptions, we show that there is a density one sequence of integers $N$ so that as $N$ tends to infinity along this sequence, all eigenfunctions of the quantized map at the inverse Planck constant $N$ are uniformly distributed. For the two-dimensional case ($g=1$), this was proved by P. Kurlberg and Z. Rudnick (2001). The higher dimensional case offers several new features and requires a completely different set of tools, including from additive combinatorics, in particular Bourgain's bound (2005) for Mordell sums, and a study of tensor product structures for the cat map.

math.DS↗

arXiv2023

On a family of sparse exponential sums

We investigate exponential sums modulo primes whose phase function is a sparse polynomial, with exponents growing with the prime. In particular, such sums model those which appear in the study of the quantum cat map. While they are not amenable to treatment by algebro-geometric methods such as Weil's bounds, Bourgain (2005) gave a nontrivial estimate for these and more general sums. In this work we obtain explicit bounds with reasonable savings over various types of averaging. We also initiate the study of the value distribution of these sums.

math.NT↗

arXiv2021

On the sum of digits of $1/M$ in $\mathbb{F}_q[x]$

For certain primes $p$, the average digit in the expansion of $1/p$ was found to have a deviation from random behaviour related to the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-p})$ (Girstmair 1994). In this short note, we observe that for the corresponding problem when we replace the integers by polynomials over a finite field, there is never any bias. The argument is elementary.

math.NT↗

arXiv2021

Differences between Robin and Neumann eigenvalues

Let $Ω\subset \mathbb R^2$ be a bounded planar domain, with piecewise smooth boundary $\partial Ω$. For $σ>0$, we consider the Robin boundary value problem \[ -Δf =λf, \qquad \frac{\partial f}{\partial n} + σf = 0 \mbox{ on } \partial Ω\] where $ \frac{\partial f}{\partial n} $ is the derivative in the direction of the outward pointing normal to $\partial Ω$. Let $0<λ^σ_0\leq λ^σ_1\leq \dots $ be the corresponding eigenvalues. The purpose of this paper is to study the Robin-Neumann gaps \[ d_n(σ):=λ_n^σ-λ_n^0 . \] For a wide class of planar domains we show that there is a limiting mean value, equal to $2{\rm length}(\partialΩ)/{\rm area}(Ω)\cdot σ$ and in the smooth case, give an upper bound of $d_n(σ)\leq C(Ω) n^{1/3}σ$ and a uniform lower bound. For ergodic billiards we show that along a density-one subsequence, the gaps converge to the mean value. We obtain further properties for rectangles, where we have a uniform upper bound, and for disks, where we improve the general upper bound.

math.AP↗

arXiv2021

Pair arithmetical equivalence for quadratic fields

Given two distinct number fields $K$ and $M$, and finite order Hecke characters $χ$ of $K$ and $η$ of $M$ respectively, we say that the pairs $(χ, K)$ and $(η, M)$ are arithmetically equivalent if the associated L-functions coincide: $$L(s, χ, K) = L(s, η, M) .$$ When the characters are trivial, this reduces to the question of fields with the same Dedekind zeta function, investigated by Gassman in 1926, who found such fields of degree 180, and by Perlis (1977) and others, who showed that there are no nonisomorphic fields of degree less than $7$. We construct infinitely many such pairs where the fields are quadratic. This gives dihedral automorphic forms induced from characters of different quadratic fields. We also give a classification of such characters of order 2 for the quadratic fields of our examples, all with odd class number.

math.NT↗

arXiv2020

A lower bound on the LCM of polynomial sequences

Let $f$ be a polynomial $f$ of degree $d\ge 2$ with integer coefficients which is irreducible over the rationals. Cilleruelo conjectured that the least common multiple of the values of the polynomial at the first $N$ integers satisfies $\log lcm(f(1),\dots, f(N)) \sim (d-1) N\log N$ as $N\to \infty$. This is only known for degree $d=2$. In this note we give a simple lower bound for all degrees $d\geq 2$ which is consistent with the conjecture: $\log lcm (f(1),\dots, f(N)) \gg N\log N$.

math.NT↗

arXiv2018

Points on nodal lines with given direction

We study of the directional distribution function of nodal lines for eigenfunctions of the Laplacian on a planar domain. This quantity counts the number of points where the normal to the nodal line points in a given direction. We give upper bounds for the flat torus, and compute the expected number for arithmetic random waves.

math.SP↗

arXiv2016

Small gaps in the spectrum of the rectangular billiard

We study the size of the minimal gap between the first N eigenvalues of the Laplacian on a rectangular billiard having irrational squared aspect ratio $α$, in comparison to the corresponding quantity for a Poissonian sequence. If $α$ is a quadratic irrationality of certain type, such as the square root of a rational number, we show that the minimal gap is roughly of size 1/N, which is essentially consistent with Poisson statistics. We also give related results for a set of $α$'s of full measure. However, on a fine scale we show that Poisson statistics is violated for all $α$. The proofs use a variety of ideas of an arithmetical nature, involving Diophantine approximation, the theory of continued fractions, and results in analytic number theory.

math.AP↗

arXiv2016

Nodal intersections for random waves on the 3-dimensional torus

We investigate the number of nodal intersections of random Gaussian Laplace eigenfunctions on the standard three-dimensional flat torus with a fixed smooth reference curve, which has nowhere vanishing curvature. The expected intersection number is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. Our main result gives a bound for the variance, if either the torsion of the curve is nowhere zero or if the curve is planar.

math.NT↗

arXiv2015

Gap Distributions in Circle Packings

We determine the distribution of nearest neighbour spacings between the tangencies to a fixed circle in a class of circle packings generated by reflections. We use a combination of geometric tools and the theory of automorphic forms.

math.NT↗

arXiv2015

Sums of divisor functions in $F_{q}[t]$ and matrix integrals

We study the mean square of sums of the $k$th divisor function $d_k(n)$ over short intervals and arithmetic progressions for the rational function field over a finite field of $q$ elements. In the limit as $q\rightarrow\infty$ we establish a relationship with a matrix integral over the unitary group. Evaluating this integral enables us to compute the mean square of the sums of $d_k(n)$ in terms of a lattice point count. This lattice point count can in turn be calculated in terms of certain polynomials, which we analyse. Our results suggest general conjectures for the corresponding classical problems over the integers, which agree with the few cases where the answer is known.

math.NT↗

arXiv2015

Some problems in analytic number theory for polynomials over a finite field

The lecture, given at the ICM 2014 in Seoul, explores several problems of analytic number theory in the context of function fields over a finite field, where they can be approached by methods different than those of traditional analytic number theory. The resulting theorems can be used to check existing conjectures over the integers, and to generate new ones. Among the problems discussed are: Counting primes in short intervals and in arithmetic progressions; Chowla's conjecture on the autocorrelation of the Mobius function; and the additive divisor problem.

math.NT↗

arXiv2014

Nodal intersections for random eigenfunctions on the torus

We investigate the number of nodal intersections of random Gaussian Laplace eigenfunctions on the standard two-dimensional flat torus ("arithmetic random waves") with a fixed real-analytic reference curve with nonvanishing curvature. The expected intersection number is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. Our main result prescribes the asymptotic behaviour of the nodal intersections variance for analytic curves in the high energy limit; remarkably, it is dependent on both the angular distribution of lattice points lying on the circle with radius corresponding to the given wavenumber, and the geometry of the given curve. In particular, this implies that the nodal intersection number admits a universal asymptotic law with arbitrarily high probability.

math-ph↗

arXiv2014

Nodal intersections and Lp restriction theorems on the torus

We study the number of intersections of the nodal lines of an eigenfunction of the Laplacian on the standard torus with a fixed reference curve, that is, the number of zeros of the eigenfunction restricted to the curve. An upper bound is the wave number k. When the curve has nowhere zero curvature, we conjecture that, up to a constant multiple, this should also be the correct a lower bound. We give a lower bound which differs from this by an arithmetic quantity, given in terms of the maximal number of lattice points in arcs of size square root of the wave number k on a circle of radius k. According to a conjecture of Cilleruelo and Granville, this quantity is bounded in which case we recover our conjecture. To get at the lower bound, we reduce the problem to giving a lower bound for the L1 norm of the restriction of the eigenfunction to the curve, and then to an upper bound for the L4 restriction norm.

math.AP↗

arXiv2013

Squarefree values of polynomials over the rational function field

We study representation of square-free polynomials in the polynomial ring F[t] over a finite field F by polynomials in F[t][x]. This is a function field version of the well-studied problem of representing squarefree integers by integer polynomials, where it is conjectured that a separable polynomial f(x) with integer coefficients takes infinitely many squarefree values, barring some simple exceptional cases, in fact that the integers n for which f(n) is squarefree have a positive density. We show that if f(x) in F[t][x] is separable, with square-free content, of bounded degree and height, then as the finite field size #F tends to infinity, for almost all monic polynomials a(t), the polynomial f(a) is squarefree.

math.NT↗