arXiv subjects

Nadav Yesha

Publications and source records attributed to Nadav Yesha.

18 recordsLinked to original sources

arXiv2025

On the number of lattice points in thin sectors

On the circle of radius $R$ centred at the origin, consider a ``thin'' sector about the fixed line $y = αx$ with edges given by the lines $y = (α\pm ε) x$, where $ε= ε_R \rightarrow 0$ as $ R \to \infty $. We establish an asymptotic count for $S_α(ε,R)$, the number of integer lattice points lying in such a sector. Our results depend both on the decay rate of $ε$ and on the rationality/irrationality type of $α$. In particular, we demonstrate that if $α$ is Diophantine, then $S_α(ε,R)$ is asymptotic to the area of the sector, so long as $εR^{t} \rightarrow \infty$ for some $ t<2 $.

math.NT↗

arXiv2025

On the supremum of random cusp forms

A random ensemble of cusp forms for the full modular group is introduced. For a weight-$k$ cusp form, restricted to a compact subdomain of the modular surface, the true order of magnitude of its expected supremum is determined to be $\asymp \sqrt{\log{k}}$, in line with the conjectured bounds. Additionally, the exponential concentration of the supremum around its median is established. Contrary to the compact case, it is shown that the global expected supremum, which is attained around the cusp, grows like $k^{1/4}$, up to a logarithmic factor.

math.NT↗

arXiv2025

Almost sure asymptotics for the number variance of dilations of integer sequences

Let $(x_n)_{n=1}^\infty$ be a sequence of integers. We study the number variance of dilations $(αx_n)_{n=1}^\infty$ modulo 1 in intervals of length $S$, and establish pseudorandom (Poissonian) behavior for Lebesgue-almost all $α$ throughout a large range of $S$, subject to certain regularity assumptions imposed upon $(x_n)_{n=1}^\infty$. For the important special case $x_n = p(n)$, where $p$ is a polynomial with integer coefficients of degree at least 2, we prove that the number variance is Poissonian for almost all $α$ throughout the range $0 \leq S \leq (\log N)^{-c}$, for a suitable absolute constant $c>0$. For more general sequences $(x_n)_{n=1}^\infty$, we give a criterion for Poissonian behavior for generic $α$ which is formulated in terms of the additive energy of the finite truncations $(x_n)_{n=1}^N$.

math.NT↗

arXiv2023

On the number variance of sequences with small additive energy

For a real-valued sequence $(x_n)_{n=1}^\infty$, denote by $S_N(\ell)$ the number of its first $N$ fractional parts lying in a random interval of size $\ell:=L/N$, where $L=o(N)$ as $N\to\infty$. We study the variance of $S_N(\ell)$ (the number variance) for sequences of the form $x_n=αa_n$, where $(a_n)_{n=1}^\infty$ is a sequence of distinct integers. We show that if the additive energy of the sequence $(a_n)_{n=1}^\infty$ is bounded from above by $N^{5/2-\varepsilon}/L$ for some $\varepsilon>0$, then for almost all $α$, the number variance is asymptotic to $L$ (Poissonian number variance). This holds in particular for the sequence $x_n=αn^d, d\ge 2$ whenever $L=N^β$ with $0\leβ<1/2$.

math.NT↗

arXiv2022

Intermediate-scale statistics for real-valued lacunary sequences

We study intermediate-scale statistics for the fractional parts of the sequence $(αa_n)_{n=1}^{\infty}$, where $(a_n)_{n=1}^{\infty}$ is a positive, real-valued lacunary sequence, and $α\in\mathbb{R}$. In particular, we consider the number of elements $S_{N}(L,α)$ in a random interval of length $L/N$, where $L=O\left(N^{1-ε}\right)$, and show that its variance (the number variance) is asymptotic to $L$ with high probability w.r.t. $α$, which is in agreement with the statistics of uniform i.i.d. random points in the unit interval. In addition, we show that the same asymptotics holds almost surely in $α\in\mathbb{R}$ when $L=O\left(N^{1/2-ε}\right)$. For slowly growing $L$, we further prove a central limit theorem for $S_{N}(L,α)$ which holds for almost all $α\in\mathbb{R}$.

math.NT↗

arXiv2021

The distribution of spacings of real-valued lacunary sequences modulo one

Let $\left(a_{n}\right)_{n=1}^{\infty}$ be a lacunary sequence of positive real numbers. Rudnick and Technau showed that for almost all $α\in\mathbb{R}$, the pair correlation of $\left(αa_{n}\right)_{n=1}^{\infty}$ mod 1 is Poissonian. We show that all higher correlations and hence the nearest-neighbour spacing distribution are Poissonian as well, thereby extending a result of Rudnick and Zaharescu to real-valued sequences.

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↗

arXiv2020

Gap statistics and higher correlations for geometric progressions modulo one

Koksma's equidistribution theorem from 1935 states that for Lebesgue almost every $α>1$, the fractional parts of the geometric progression $(α^{n})_{n\geq1}$ are equidistributed modulo one. In the present paper we sharpen this result by showing that for almost every $α>1$, the correlations of all finite orders and hence the normalized gaps of $(α^{n})_{n\geq1}$ mod 1 have a Poissonian limit distribution, thereby resolving a conjecture of the two first named authors. While an earlier approach used probabilistic methods in the form of martingale approximation, our reasoning in the present paper is of an analytic nature and based upon the estimation of oscillatory integrals. This method is robust enough to allow us to extend our results to a natural class of sub-lacunary sequences.

math.NT↗

arXiv2020

On the correlations of $n^α$ mod 1

A well known result in the theory of uniform distribution modulo one (which goes back to Fejér and Csillag) states that the fractional parts $\{n^α\}$ of the sequence $(n^α)_{n\ge1}$ are uniformly distributed in the unit interval whenever $α>0$ is not an integer. For sharpening this knowledge to local statistics, the $k$-level correlation functions of the sequence $(\{n^α\})_{n\geq1}$ are of fundamental importance. We prove that for each $k\ge2,$ the $k$-level correlation function $R_k$ is Poissonian for almost every $α>4k^2-4k-1$.

math.NT↗

arXiv2020

The defect of toral Laplace eigenfunctions and Arithmetic Random Waves

We study the defect (or "signed area") distribution of toral Laplace eigenfunctions restricted to shrinking balls of radius above the Planck scale, in either random Gaussian scenario ("Arithmetic Random Waves"), or deterministic eigenfunctions averaged w.r.t. the spatial variable. In either scenario we exploit the associated symmetry of the eigenfunctions to show that the expectation (Gaussian or spatial) vanishes. Our principal results concern the high energy limit behaviour of the defect variance.

math-ph↗

arXiv2019

Small scale equidistribution for a point scatterer on the torus

We study the small scale distribution of the eigenfunctions of a point scatterer (the Laplacian perturbed by a delta potential) on two- and three-dimensional flat tori. In two dimensions, we establish small scale equidistribution for the "new" eigenfunctions holding all the way down to the Planck scale. In three dimensions, small scale equidistribution is established for all of the "new" eigenfunctions at certain scales.

math-ph↗

arXiv2019

CLT for Planck scale mass distribution of toral Laplace eigenfunctions

We study the fine scale $L^2$-mass distribution of toral Laplace eigenfunctions with respect to random position, in 2 and 3 dimensions. In 2d, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy levels, both the asymptotic shape of the variance is determined and the limiting Gaussian law is established, in the optimal Planck-scale regime. In 3d the asymptotic behaviour of the variance is analysed in a more restrictive scenario ("Bourgain's eigenfunctions"). Other than the said precise results, lower and upper bounds are proved for the variance, under more general flatness assumptions on the Fourier coefficients.

math.NT↗

arXiv2017

Pair correlation for quadratic polynomials mod 1

It is an open question whether the fractional parts of nonlinear polynomials at integers have the same fine-scale statistics as a Poisson point process. Most results towards an affirmative answer have so far been restricted to almost sure convergence in the space of polynomials of a given degree. We will here provide explicit Diophantine conditions on the coefficients of polynomials of degree 2, under which the convergence of an averaged pair correlation density can be established. The limit is consistent with the Poisson distribution. Since quadratic polynomials at integers represent the energy levels of a class of integrable quantum systems, our findings provide further evidence for the Berry-Tabor conjecture in the theory of quantum chaos.

math.NT↗

arXiv2017

Uniform Distribution of Eigenstates on a Torus with Two Point Scatterers

We study the Laplacian perturbed by two delta potentials on a two-dimensional flat torus. There are two types of eigenfunctions for this operator: old, or unperturbed eigenfunctions which are eigenfunctions of the standard Laplacian, and new, perturbed eigenfunctions which are affected by the scatterers. We prove that along a density one sequence, the new eigenfunctions are uniformly distributed in configuration space, provided that the difference of the scattering points is Diophantine.

math-ph↗

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↗

arXiv2014

On the distribution of the divisor function and Hecke eigenvalues

We investigate the behavior of the divisor function in both short intervals and in arithmetic progressions. The latter problem was recently studied by É. Fouvry, S. Ganguly, E. Kowalski, and Ph. Michel. We prove a complementary result to their main theorem. We also show that in short intervals of certain lengths the divisor function has a Gaussian limiting distribution. The analogous problems for Hecke eigenvalues are also considered.

math.NT↗

arXiv2013

Quantum ergodicity for a point scatterer on the three-dimensional torus

Consider a point scatterer (the Laplacian perturbed by a delta-potential) on the standard three-dimensional flat torus. Together with the eigenfunctions of the Laplacian which vanish at the point, this operator has a set of new, perturbed eigenfunctions. In a recent paper, the author was able to show that all of the perturbed eigenfunctions are uniformly distributed in configuration space. In this paper we prove that almost all of these eigenfunctions are uniformly distributed in phase space, i.e. we prove quantum ergodicity for the subspace of the perturbed eigenfunctions. An analogue result for a point scatterer on the two-dimensional torus was recently proved by Kurlberg and Ueberschär.

math.AP↗

arXiv2012

Eigenfunction statistics for a point scatterer on a three-dimensional torus

In this paper we study eigenfunction statistics for a point scatterer (the Laplacian perturbed by a delta-potential) on a three-dimensional flat torus. The eigenfunctions of this operator are the eigenfunctions of the Laplacian which vanish at the scatterer, together with a set of new eigenfunctions (perturbed eigenfunctions). We first show that for a point scatterer on the standard torus all of the perturbed eigenfunctions are uniformly distributed in configuration space. Then we investigate the same problem for a point scatterer on a flat torus with some irrationality conditions, and show uniform distribution in configuration space for almost all of the perturbed eigenfunctions.

math.AP↗