Related papers: The Gauss circle problem for Penrose tilings

Topology and geometry are deeply intertwined in the study of surfaces, though their interaction manifests differently in smooth and discrete settings. In the smooth category, a classical result asserts that any closed smooth surface…

We consider the system of $N$ ($\ge2$) hard balls with masses $m_1,...,m_N$ and radius $r$ in the flat torus $\Bbb T_L^\nu=\Bbb R^\nu/L\cdot\Bbb Z^\nu$ of size $L$, $\nu\ge3$. We prove the ergodicity (actually, the Bernoulli mixing…

Let R be a complete discrete valuation ring of mixed characteristics, with algebraically closed residue field k. We study the existence problem of equivariant liftings to R of Galois covers of nodal curves over k. Using formal geometry, we…

In this article, we continue the classification of finite type Gauss map surfaces in the Euclidean 3-space E3 with respect to the first fundamental form by studying a subclass of tubes, namely the anchor rings. We show that anchor rings are…

Let G be a complex semi-simple group, X a Riemann surface, M_G the moduli space of principal G-bundles on X. When G is simply-connected, there exists a closed formula expressing the dimension of the space H^0(M_G,L) for any line bundle L on…

Any stretching of Ringel's non-Pappus pseudoline arrangement when projected into the Euclidean plane, implicitly contains a particular arrangement of nine triangles. This arrangement has a complex constraint involving the sines of its…

We show that every tiling of a convex set in the Euclidean plane $\mathbb{R}^2$ by equilateral triangles of mutually different sizes contains arbitrarily small tiles. The proof is purely elementary up to the discussion of one family of…

The Thurston norm of a 3-manifold measures the complexity of surfaces representing two-dimensional homology classes. We study the possible unit balls of Thurston norms of 3-manifolds $M$ with $b_1(M) = 2$, and whose fundamental groups admit…

We prove the weight part of Serre's conjecture in generic situations for forms of $U(3)$ which are compact at infinity and split at places dividing $p$ as conjectured by Herzig. We also prove automorphy lifting theorems in dimension three.…

Let V be a plane smooth cubic curve over a finitely generated field k. The Mordell-Weil theorem for V states that there is a finite subset P \subset V(k) such that the whole V(k) can be obtained from P by drawing secants and tangents…

Consider a $p$-random subset $A$ of initially infected vertices in the discrete cube $[L]^3$, and assume that the neighbourhood of each vertex consists of the $a_i$ nearest neighbours in the $\pm e_i$-directions for each $i \in \{1,2,3\}$,…

A radial probability measure is a probability measure with a density (with respect to the Lebesgue measure) which depends only on the distances to the origin. Consider the Euclidean space enhanced with a radial probability measure. A…

A genus-1 tangle G is an arc properly embedded in a standardly embedded solid torus S in the 3-sphere. We say that a genus-1 tangle embeds in a knot K in S^3 if the tangle can be completed by adding an arc exterior to the solid torus to…

We compute the number of all rhombus tilings of a hexagon with sides $a,b+1,c,a+1,b,c+1$, of which the central triangle is removed, provided $a,b,c$ have the same parity. The result is a product of four numbers, each of which counts the…

We study rays in von Mangoldt planes, which has applications to the structure of open complete manifolds with lower radial curvature bounds. We prove that the set of souls of any rotationally symmetric plane of nonnegative curvature is a…

We present a self-contained proof of the Gauss-Bonnet theorem for two-dimensional surfaces embedded in $R^3$ using just classical vector calculus. The exposition should be accessible to advanced undergraduate and non-expert graduate…

We show that a square-tiling of a $p\times q$ rectangle, where $p$ and $q$ are relatively prime integers, has at least $\log_2p$ squares. If $q>p$ we construct a square-tiling with less than $q/p+C\log p$ squares of integer size, for some…

In the present paper we discuss four ways of looking at rhombile tilings: stacking 3-dimensional cubes, elements of groups, and configurations of lines and points.

We analytically study bond percolation on hyperbolic lattices obtained by tiling a hyperbolic plane with constant negative Gaussian curvature. The quantity of our main concern is $p_{c2}$, the value of occupation probability where a unique…

In this note we present the solution of some isoperimetric problems in open convex cones of $\R^n$ in which perimeter and volume are measured with respect to certain nonradial weights. Surprisingly, Euclidean balls centered at the origin…