Related papers: The distribution of lattice points in elliptic ann…
We study the statistical properties of the counting function of lattice points inside thin annuli. By a conjecture of Bleher and Lebowitz, if the width shrinks to zero, but the area converges to infinity, the distribution converges to the…
We show that the number of lattice points lying in a thin annulus has a Gaussian value distribution if the width of the annulus tends to zero sufficiently slowly as we increase the inner radius.
For fixed coprime polynomials $U,V \in \mathbb{F}_q [T]$ with degrees of different parities, and a general polynomial $A \in \mathbb{F}_q [T]$, define the arithmetic function $S_{U,V} (A)$ to be the number of representations of $A$ of the…
We asymptotically estimate the variance of the number of lattice points in a thin, randomly rotated annulus lying on the surface of the sphere. This partially resolves a conjecture of Bourgain, Rudnick, and Sarnak. We also obtain estimates…
We evaluate the mean square limit of exponential sums related with a rational ellipsoid, extending a work of Marklof. Moreover, as a result of it, we study the asymptotic values of the normalized deviations of the number of lattice points…
We study the fine distribution of lattice points lying on expanding circles in the hyperbolic plane $\mathbb{H}$. The angles of lattice points arising from the orbit of the modular group $PSL_{2}(\mathbb{Z})$, and lying on hyperbolic…
We investigate the statistics of the number $N(R,S)$ of lattice points, $n\in \Z^2$, in a ``random'' annular domain $\Pi(R,w)=\,(R+w)A\,\setminus RA$, where $R,w >0$. Here $A$ is a fixed convex set with smooth boundary and $w$ is chosen so…
Let $\pi : E\to B$ be an elliptic surface defined over a number field $K$, where $B$ is a smooth projective curve, and let $P: B \to E$ be a section defined over $K$ with canonical height $\hat{h}_E(P)\not=0$. In this article, we show that…
The disks of spiral galaxies are generally elliptical rather than circular. The distribution of ellipticities can be fit with a log-normal distribution. For a sample of 12,764 galaxies from the Sloan Digital Sky Survey Data Release 1 (SDSS…
We give asymptotic estimates of the variance of the number of integer points in translated thin annuli in any dimension.
Given an elliptic curve $E$ and a point $P$ in $E(\mathbb{R})$, we investigate the distribution of the points $nP$ as $n$ varies over the integers, giving bounds on the $x$ and $y$ coordinates of $nP$ and determining the natural density of…
In this paper we derive an exact full expression for the 2D probability distribution of the ellipticity of an object measured from data, only assuming Gaussian noise in pixel values. This is a generalisation of the probability distribution…
The difference between the number of lattice points in a disk of radius $\sqrt{t}/2\pi$ and the area of the disk $t/4\pi$ is equal to the error in the Weyl asymptotic estimate for the eigenvalue counting function of the Laplacian on the…
On the circle of radius $R$ centred at the origin, consider a ``thin'' sector about the fixed line $y = \alpha x$ with edges given by the lines $y = (\alpha \pm \epsilon) x$, where $\epsilon = \epsilon_R \rightarrow 0$ as $ R \to \infty $.…
We show that for any $\alpha\in (1/2,1)$ the number of lattice points belonging to an arc of length $R^{\alpha}$ of the circle of radius $R$ centered at the origin is not uniformly bounded in $R$, which disproves the corresponding…
This paper is inspired by Richards' work on large gaps between sums of two squares [10]. It is shown in [10] that there exist arbitrarily large values of $\lambda$ and $\mu$, where $\mu \geq C \log \lambda$, such that intervals $[\lambda,…
An effective estimate for the lattice point discrepancy of ellipsoids of rotation. This paper provides an explicit bound, with numerical constants, for the lattice point discrepancy (= number of integer points minus volume) of an ellipsoid…
This paper presents an alternative formulation of the geodesic normal distribution on the sphere, building on the work of Hauberg (2018). While the isotropic version of this distribution is naturally defined on the sphere, the anisotropic…
We study the hexagonal lattice $\mathbb{Z}[\omega]$, where $\omega^6=1$. More specifically, we study the angular distribution of hexagonal lattice points on circles with a fixed radius. We prove that the angles are equidistributed on…
In a quadruply imaged lens system the angular distribution of images around the lens center is completely described by three relative angles. We show empirically that in the 3D space of these angles, spanning 180 x 180 x 90 degrees, quads…