Related papers: Tiling of rectangles with squares via Diophantine …
In 1996 N. Chevallier proved a beautiful lemma which connects Diophantine approximation and multidimensional generalizations of the famous Three Distance Theorem. Using this lemma we show how known results about multidimensional three…
A tiling is a decomposition of a polygon into finitely many non-overlapping triangles. We prove that if a regular n-gon, $n \geq 5$, $n \neq 28$, can be tiled with similar right triangles, then one of the angles of these triangles is in…
We present in this article a general approach (in the form of recommendations and guidelines) for tackling Diophantine equation problems (whether single equations or systems of simultaneous equations). The article should be useful in…
We completely classify edge-to-edge tilings of the sphere by congruent quadrilaterals. As part of the classification, we also present a modern version of the classification of edge-to-edge tilings of the sphere by congruent triangles.…
This paper is motivated by two problems in the theory of Diophantine approximation, namely, Davenport's problem regarding badly approximable points on submanifolds of a Euclidean space and Schmidt's problem regarding the intersections of…
We study some problems in metric Diophantine approximation over local fields of positive characteristic.
Given a finite collection of two-dimensional tile types, the field of study concerned with covering the plane with tiles of these types exclusively has a long history, having enjoyed great prominence in the last six to seven decades. Much…
We survey classical and recent results on exponents of Diophantine approximation. We give only a few proofs and highlight several open problems.
In this paper we present methods for triangulation of infinite cylinders from image line silhouettes. We show numerically that linear estimation of a general quadric surface is inherently a badly posed problem. Instead we propose to…
Does a given a set of polyominoes tile some rectangle? We show that this problem is undecidable. In a different direction, we also consider tiling a cofinite subset of the plane. The tileability is undecidable for many variants of this…
We show that given any tiling of Euclidean space, any geometric patterns of points, we can find a patch of tiles (of arbitrarily large size) so that copies of this patch appear in the tiling nearly centered on a scaled and translated…
We develop the metric theory of Diophantine approximation on homogeneous varieties of semisimple algebraic groups and prove results analogous to the classical Khinchin and Jarnik theorems. In full generality our results establish…
We consider the following geometric optimization problem: Given $ n $ axis-aligned rectangles in the plane, the goal is to find a set of horizontal segments of minimum total length such that each rectangle is stabbed. A segment stabs a…
The Hausdorff dimension of an exceptional set of periods for which convergence of a formal solution to an inhomogeneous wave equation in n spatial and one temporal dimension is problematic, is determined along with conditions which the…
This article examines the tilings of a strip with equilateral triangles. The number of ways in which the lattices can be covered with a combination of tiles of the two types of triangles is related to Pell's numbers. Additionally, the…
We develop a multifractal random tilling that fills the square. The multifractal is formed by an arrangement of rectangular blocks of different sizes, areas and number of neighbors. The overall feature of the tilling is an heterogeneous and…
The tiling problem has been a famous problem that has appeared in many Mathematics problems. Many of its solutions are rooted in high-level Mathematics. Thus we hope to tackle this problem using more elementary Mathematics concepts. In this…
The edge-to-edge tilings of the sphere by congruent polygons, where all edges are straight, have been completely classified. We classify the curvilinear version of the similar triangular tilings, where the edges may not be straight, and…
We show that a rectangle triangle random tiling with a tenfold symmetric phase is solvable by Bethe Ansatz. After the twelvefold square triangle and the eightfold rectangle triangle random tiling, this is the third example of a rectangle…
Recent developments in the theory and application of the Hardy-Littlewood method are discussed, concentrating on aspects associated with diagonal diophantine problems. Recent efficient differencing methods for estimating mean values of…