Related papers: Estimates on Lattice Points in the Circle
We use the finite lattice method to calculate the radius of gyration, the first and second area-weighted moments of self-avoiding polygons on the square lattice. The series have been calculated for polygons up to perimeter 82. Analysis of…
We show that, if the interior of a lattice d-polytope P contains at least one lattice point, then it contains a lattice point whose coefficient of asymmetry with respect to P is at most b for some number b depending on d only. As an…
A large luminosity--linewidth template sample is now available, improved absorption corrections have been derived, and there are a statistically significant number of galaxies with well determined distances to supply the zero point. A…
We begin by revisiting a paper of Erd\H{o}s and Fishburn, which posed the following question: given $k\in \mathbb{N}$, what is the maximum number of points in a plane that determine at most $k$ distinct distances, and can such optimal…
We explore the use of correlation with simple functions to get lower bounds for arithmetic quantities. In particular, we apply this idea to the power moments of the error term when counting visible lattice points in large spheres.
A circle $C$ separates two planar sets if it encloses one of the sets and its open interior disk does not meet the other set. A separating circle is a largest one if it cannot be locally increased while still separating the two given sets.…
We prove that any $n$ points in $\mathbb{R}^2$, not all on a line or circle, determine at least $\frac{1}{4}n^2-O(n)$ ordinary circles (circles containing exactly three of the $n$ points). The main term of this bound is best possible for…
We prove various estimates for the mean square lattice point discrepancy for dilates of a convex body.
Lattice radial quantization is introduced as a nonperturbative method intended to numerically solve Euclidean conformal field theories that can be realized as fixed points of known Lagrangians. As an example, we employ a lattice shaped as a…
We investigate the average number of lattice points within a ball for the $n$th cyclotomic number field, where the lattice is chosen at random from the set of unit determinant ideal lattices of the field. We show that this average is nearly…
Let $n$ be a positive integer. In this paper we estimate the size of the set of linear forms $b_1\log a_1 + b_2\log a_2+...+b_n\log a_n$, where $|b_i|\leq B_i$ and $1\leq a_i\leq A_i$ are integers, as $A_i,B_i\to \infty$.
We develop a new approach to address some classical questions concerning the size and structure of integer distance sets. Our main result is that any integer distance set in the Euclidean plane is either very sparse or has all but an…
Let $\mathcal{S}$ be a finite set of integer points in $\mathbb{R}^d$, which we assume has many symmetries, and let $P\in\mathbb{R}^d$ be a fixed point. We calculate the distances from $P$ to the points in $\mathcal{S}$ and compare the…
Let $\mathcal A$ be a star-shaped polygon in the plane, with rational vertices, containing the origin. The number of primitive lattice points in the dilate $t\mathcal A$ is asymptotically $\frac6{\pi^2}$ Area$(t\mathcal A)$ as $t\to…
Two polygons are amicable if the perimeter of one is equal to the area of the other and vice versa. A polygon is a lattice polygon if its vertices are on the integer lattice $\Z^2$. We show that there is one pair of amicable lattice…
In this paper, we continue the study of three close factorizations of an integer and correct a mistake of a previous result. This turns out to be related to lattice points close to the center point $(\sqrt{N}, \sqrt{N})$ of the hyperbola $x…
We study ideal lattices in $\mathbb{R}^2$ coming from real quadratic fields, and give an explicit method for computing all well-rounded twists of any such ideal lattice. We apply this to ideal lattices coming from Markoff numbers to…
We study a lattice point counting problem for spheres arising from the Heisenberg groups. In particular, we prove an upper bound on the number of points on and near large dilates of the unit spheres generated by the anisotropic norms…
In this note, we derive an asymptotically sharp upper bound on the number of lattice points in terms of the volume of centrally symmetric convex bodies. Our main tool is a generalization of a result of Davenport that bounds the number of…
We estimate the $L^{p}$ norms of the discrepancy between the volume and the number of integer points in $r\Omega-x$, a dilated by a factor $r$ and translated by a vector $x$ of a convex body $\Omega$ in $\mathbb{R}^{d}$ with smooth boundary…