Related papers: Some Problems in Number Theory I: The Circle Probl…
This paper provides estimates on the difference between the number of integer lattice points an a circle centered at the origin and the area. The estimates have the form "Big O" of the product of logarithm of the radius and the radius…
The Gauss Circle Problem concerns finding asymptotics for the number of lattice point lying inside a circle in terms of the radius of the circle. The heuristic that the number of points is very nearly the area of the circle is surprisingly…
In this paper, we propose a class of elementary plane geometry problems closely related to the title of this paper. Here, a circle is the 1-dimensional curve bounding a disk. For any nonnegative integer, a circle is called $n$-enclosing if…
This PhD thesis deals with a number of different problems in mathematical physics with the common thread that they have probabilistic aspects. The problems all stem from mathematical studies of lattice systems in statistical and quantum…
This is a survey article on the theory of lattice points in large planar domains and bodies of dimensions 3 and higher, with an emphasis on recent developments and new methods, including a lot of results established only during the last few…
Given a circle of radius $r$ centered at the origin, the Gauss Circle Problem concerns counting the number of lattice points $C(r)$ within this circle. It is known that as $r$ grows large, the number of lattice points approaches $\pi r^2$,…
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…
The distribution of lattice points with relatively $r$-prime is related to problems in the Number Theory such as the Extended Lindel\"{o}f Hypothesis and the Gauss Circle Problem. It is known that Sittinger's result is improved on the…
Rudin conjectured that there are never more than c N^(1/2) squares in an arithmetic progression of length N. Motivated by this surprisingly difficult problem we formulate more than twenty conjectures in harmonic analysis, analytic number…
We study lattice points in d-dimensional spheres, and count their number in thin spherical segments. We found an upper bound depending only on the radius of the sphere and opening angle of the segment. To obtain this bound we slice the…
The number of lattice points $\left| tP \cap \mathbb{Z}^d \right|$, as a function of the real variable $t>1$ is studied, where $P \subset \mathbb{R}^d$ belongs to a special class of algebraic cross-polytopes and simplices. It is shown that…
We study lattice points on hyperbolic circles centred at Heegner points of class number one. Our main result is that, on a density one subset of radii tending to infinity, the angles of such points equidistribute on the unit circle. To…
A problem that is simple to state in the context of spherical geometry, and that seems rather interesting, appears to have been unexamined to date in the mathematical literature. The problem can also be recast as a problem in the real…
This paper is concerned with configurations of points in a plane lattice which determine angles that are rational multiples of $\pi$. We shall study how many such angles may appear in a given lattice and in which positions, allowing the…
This paper discusses a problem that consists of $n$ "lighthouses" which are circles with radius 1, placed around a common center, equidistant at $n$ units away from the placement center. Consecutive lighthouses are separated by the same…
We present five open problems in the theory of vertex rings. They cover a variety of different areas of research where vertex rings have been, or are threatening to be, relevant. They have also been chosen because I personally find them…
Sources of uncertainties in perturbative calculations, tadpole improvement and its role in lattice perturbation theory, and six recent calculations are discussed.
This is a pedagogical article cited in the foregoing research note, quant-ph/9911050
This paper concerns the number of lattice points in the plane which are visible along certain curves to all elements in some set S of lattice points simultaneously. By proposing the concept of level of visibility, we are able to analyze…
This book has four chapters. Chapter one is introductory in nature, for it recalls some basic definitions essential to make the book a self-contained one. Chapter two, introduces for the first time the new notion of neutrosophic rings and…