Structure Results for Multiple Tilings in 3D

We study multiple tilings of 3-dimensional Euclidean space by a convex body. In a multiple tiling, a convex body $P$ is translated with a discrete multiset $\Lambda $ in such a way that each point of ${\mathbb {R}}^d$ gets covered exactly $k$ times, except perhaps the translated copies of the boundary of $P$ . It is known that all possible multiple tilers in ${\mathbb {R}}^3$ are zonotopes. In ${\mathbb {R}}^2$ it was known by the work of Kolountzakis (Discrete Comput Geom 23(4):537–553, 2000) that, unless $P$ is a parallelogram, the multiset of translation vectors $\Lambda $ must be a finite union of translated lattices (also known as quasi periodic sets). In that work (Kolountzakis, Discrete Comput Geom 23(4):537–553, 2000) the author asked whether the same quasi-periodic structure on the translation vectors would be true in ${\mathbb {R}}^3$ . Here we prove that this conclusion is indeed true for ${\mathbb {R}}^3$ . Namely, we show that if $P$ is a convex multiple tiler in ${\mathbb {R}}^3$ , with a discrete multiset $\Lambda $ of translation vectors, then $\Lambda $ has to be a finite union of translated lattices, unless $P$ belongs to a special class of zonotopes. This exceptional class consists of two-flat zonotopes $P$ , defined by the Minkowski sum of two 2-dimensional symmetric polygons in ${\mathbb {R}}^3$ , one of which may degenerate into a single line segment. It turns out that rational two-flat zonotopes admit a multiple tiling with an aperiodic (nonquasi-periodic) set of translation vectors $\Lambda $ . We note that it may be quite difficult to offer a visualization of these 3-dimensional non-quasi-periodic tilings, and that we discovered them by using Fourier methods.     

Metrics on tiling spaces, local isomorphism and an application of Brown’s lemma

We give an application of a topological dynamics version of multidimensional Brown’s lemma to tiling theory: given a tiling of an Euclidean space and a finite geometric pattern of points $F$ , one can find a patch such that, for each scale factor $\lambda $ , there is a vector $\vec {t}_\lambda $ so that copies of this patch appear in the tilling “nearly” centered on $\lambda F+\vec {t}_\lambda $ once we allow “bounded perturbations” in the structure of the homothetic copies of $F$ . Furthermore, we introduce a new unifying setting for the study of tiling spaces which allows rather general group “actions” on patches and we discuss the local isomorphism property of tilings within this setting.     

Tiling Groupoids and Bratteli Diagrams

Let T be an aperiodic and repetitive tiling of ${{\mathbb R}^d}$ with finite local complexity. Let Ω be its tiling space with canonical transversal ${\Xi}$ . The tiling equivalence relation ${R_\Xi}$ is the set of pairs of tilings in ${\Xi}$ which are translates of each others, with a certain (étale) topology. In this paper ${R_\Xi}$ is reconstructed as a generalized “tail equivalence” on a Bratteli diagram, with its standard AF -relation as a subequivalence relation. Using a generalization of the Anderson–Putnam complex (Bellissard et al. in Commun. Math. Phys. 261:1–41, 2006) Ω is identified with the inverse limit of a sequence of finite CW-complexes. A Bratteli diagram ${{\mathcal B}}$ is built from this sequence, and its set of infinite paths ${\partial {\mathcal B}}$ is homeomorphic to ${\Xi}$ . The diagram ${{\mathcal B}}$ is endowed with a horizontal structure: additional edges that encode the adjacencies of patches in T. This allows to define an étale equivalence relation ${R_{\mathcal B}}$ on ${\partial {\mathcal B}}$ which is homeomorphic to ${R_\Xi}$ , and contains the AF-relation of “tail equivalence”.     

Collapsibility and Vanishing of Top Homology in Random Simplicial Complexes

Let $\Delta _{n-1}$ denote the $(n-1)$ -dimensional simplex. Let $Y$ be a random $d$ -dimensional subcomplex of $\Delta _{n-1}$ obtained by starting with the full $(d-1)$ -dimensional skeleton of $\Delta _{n-1}$ and then adding each $d$ -simplex independently with probability $p=\frac{c}{n}$ . We compute an explicit constant $\gamma _d$ , with $\gamma _2 \simeq 2.45$ , $\gamma _3 \simeq 3.5$ , and $\gamma _d=\Theta (\log d)$ as $d \rightarrow \infty $ , so that for $c < \gamma _d$ such a random simplicial complex either collapses to a $(d-1)$ -dimensional subcomplex or it contains $\partial \Delta _{d+1}$ , the boundary of a $(d+1)$ -dimensional simplex. We conjecture this bound to be sharp. In addition, we show that there exists a constant $\gamma _d< c_d <d+1$ such that for any $c>c_d$ and a fixed field $\mathbb{F }$ , asymptotically almost surely $H_d(Y;\mathbb{F }) \ne 0$ .     

Tilings of Parallelograms with Similar Right Triangles

We say that a triangle $T$ T tiles the polygon $\mathcal A $ A if $\mathcal A $ A can be decomposed into finitely many non-overlapping triangles similar to $T$ T . A tiling is called regular if there are two angles of the triangles, say $\alpha $ α and $\beta $ β , such that at each vertex $V$ V of the tiling the number of triangles having $V$ V as a vertex and having angle $\alpha $ α at $V$ V is the same as the number of triangles having angle $\beta $ β at $V$ V . Otherwise the tiling is called irregular. Let $\mathcal P (\delta )$ P ( δ ) be a parallelogram with acute angle $\delta $ δ . In this paper we prove that if the parallelogram $\mathcal P (\delta )$ P ( δ ) is tiled with similar triangles of angles $(\alpha , \beta , \pi /2)$ ( α , β , π / 2 ) , then $(\alpha , \beta )=(\delta , \pi /2-\delta )$ ( α , β ) = ( δ , π / 2 - δ ) or $(\alpha , \beta )=(\delta /2, \pi /2-\delta /2)$ ( α , β ) = ( δ / 2 , π / 2 - δ / 2 ) , and if the tiling is regular, then only the first case can occur.     

Helly-type theorems for intersections of sets starshaped via orthogonally convex paths

Let ${\mathcal{K}}$ be a family of simply connected sets in the plane. If every countable subfamily of ${\mathcal{K}}$ has an intersection that is starshaped via orthogonally convex paths, then ${\mathcal{K}}$ itself has such an intersection. For the d-dimensional case, let ${\mathcal{K}}$ be a family of compact sets in ${\mathbb{R}^d}$ . If every finite subfamily of ${\mathcal{K}}$ has an intersection that is starshaped via orthogonally convex paths, again ${\mathcal{K}}$ itself has such an intersection.     

Ricci flow of homogeneous manifolds

We present in this paper a general approach to study the Ricci flow on homogeneous manifolds. Our main tool is a dynamical system defined on a subset $\mathcal H _{q,n}$ of the variety of $(q+n)$ -dimensional Lie algebras, parameterizing the space of all simply connected homogeneous spaces of dimension $n$ with a $q$ -dimensional isotropy, which is proved to be equivalent in a precise sense to the Ricci flow. The approach is useful to better visualize the possible (nonflat) pointed limits of Ricci flow solutions, under diverse rescalings, as well as to determine the type of the possible singularities. Ancient solutions arise naturally from the qualitative analysis of the evolution equation. We develop two examples in detail: a $2$ -parameter subspace of $\mathcal H _{1,3}$ reaching most of $3$ -dimensional geometries, and a $2$ -parameter family in $\mathcal H _{0,n}$ of left-invariant metrics on $n$ -dimensional compact and non-compact semisimple Lie groups.     

Variance asymptotics for random polytopes in smooth convex bodies

Let $K \subset \mathbb R ^d$ be a smooth convex set and let $\mathcal{P }_{\lambda }$ be a Poisson point process on $\mathbb R ^d$ of intensity ${\lambda }$ . The convex hull of $\mathcal{P }_{\lambda }\cap K$ is a random convex polytope $K_{\lambda }$ . As ${\lambda }\rightarrow \infty $ , we show that the variance of the number of $k$ -dimensional faces of $K_{\lambda }$ , when properly scaled, converges to a scalar multiple of the affine surface area of $K$ . Similar asymptotics hold for the variance of the number of $k$ -dimensional faces for the convex hull of a binomial process in $K$ .     

Existence of integral m-varifolds minimizing \int \!|A|^p and \int \!|H|^p,\,p>m, in Riemannian manifolds

We prove existence of integral rectifiable $m$ -dimensional varifolds minimizing functionals of the type $\int |H|^p$ and $\int |A|^p$ in a given Riemannian $n$ -dimensional manifold $(N,g),\,2\le m<n$ and $p>m,$ under suitable assumptions on $N$ (in the end of the paper we give many examples of such ambient manifolds). To this aim we introduce the following new tools: some monotonicity formulas for varifolds in ${\mathbb{R }^S}$ involving $\int |H|^p,$ to avoid degeneracy of the minimizer, and a sort of isoperimetric inequality to bound the mass in terms of the mentioned functionals.     

Global symplectic coordinates on gradient Kähler–Ricci solitons

A classical result of McDuff [14] asserts that a simply connected complete Kähler manifold $(M,g,\omega )$ with non positive sectional curvature admits global symplectic coordinates through a symplectomorphism $\Psi \ : M \rightarrow \mathbb{R }^{2n}$ (where $n$ is the complex dimension of $M$ ), satisfying the following property (proved by E. Ciriza in [4]): the image $\Psi (T)$ of any complex totally geodesic submanifold $T\subset M$ through the point $p$ such that $\Psi (p)=0$ , is a complex linear subspace of $\mathbb C ^n\simeq \mathbb{R }^{2n}$ . The aim of this paper is to exhibit, for all positive integers $n$ , examples of $n$ -dimensional complete Kähler manifolds with non-negative sectional curvature globally symplectomorphic to $\mathbb{R }^{2n}$ through a symplectomorphism satisfying Ciriza’s property.     

A construction of cylindric and polyadic algebras from atomic relation algebras

Given a simple atomic relation algebra ${\mathcal{A}}$ and a finite n ?? 3, we construct effectively an atomic n-dimensional polyadic equality-type algebra ${\mathcal{P}}$ such that for any subsignature L of the signature of ${\mathcal{P}}$ that contains the boolean operations and cylindrifications, the L-reduct of ${\mathcal{P}}$ is completely representable if and only if ${\mathcal{A}}$ is completely representable. If ${\mathcal{A}}$ is finite then so is ${\mathcal{P}}$ . It follows that there is no algorithm to determine whether a finite n-dimensional cylindric algebra, diagonal-free cylindric algebra, polyadic algebra, or polyadic equality algebra is representable (for diagonal-free algebras this was known). We also obtain a new proof that the classes of completely representable n-dimensional algebras of these types are non-elementary, a result that remains true for infinite dimensions if the diagonals are present, and also for infinite-dimensional diagonal-free cylindric algebras.     

Hausdorff measures of subgroups of \mathbb{R }/\mathbb{Z } and \mathbb{R }

For a sequence $\underline{u}=(u_n)_{n\in \mathbb{N }}$ of integers, let $t_{\underline{u}}(\mathbb{T })$ be the group of all topologically $\underline{u}$ -torsion elements of the circle group $\mathbb{T }:=\mathbb{R }/\mathbb{Z }$ . We show that for any $s\in ]0,1[$ and $m\in \{0,+\infty \}$ there exists $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has Hausdorff dimension $s$ and $s$ -dimensional Hausdorff measure equal to $m$ (no other values for $m$ are possible). More generally, for dimension functions $f,g$ with $f(t)\prec g(t), f(t)\prec \!\!\!\prec t$ and $g(t)\prec \!\!\!\prec t$ we find $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has at the same time infinite $f$ -measure and null $g$ -measure.     

Regular prism tilings in {{\widetilde{\bf SL_{2}R}}} space

\({{\widetilde{\bf SL_{2}R}}}\) geometry is one of the eight 3-dimensional Thurston geometries, it can be derived from the 3-dimensional Lie group of all 2 × 2 real matrices with determinant one. Our aim is to describe and visualize the regular infinite or bounded p-gonal prism tilings in \({{\widetilde{\bf SL_{2}R}}}\) . For this purpose we introduce the notion of infinite and bounded prisms, prove that there exist infinitely many regular infinite p-gonal face-to-face prism tilings \({\mathcal{T}^i_p(q)}\) and infinitely many regular bounded p-gonal non-face-to-face \({{\widetilde{\bf SL_{2}R}}}\) prism tilings \({\mathcal{T}_p(q)}\) for integer parameters \({p,q; 3 \leq p, \frac{2p}{p-2} < q}\) . Moreover, we describe the symmetry group of \({\mathcal{T}_p(q)}\) via its index 2 rotational subgroup, denoted by pq2 ₁ . Surprisingly this group already occurred in our former work (Molnár et al., J Geometry, 95:91–133, 2009) in another context. We also develop a method to determine the data of the space filling regular infinite and bounded prism tilings. We apply the above procedure to \({\mathcal{T}^i_3(q)}\) and \({\mathcal{T}_3(q)}\) where 6 < q and visualize them and the corresponding tilings. E. Molnár showed, that homogeneous 3-spaces have a unified interpretation in the projective 3-sphere \({\mathcal{PS}^3}\) and 3-space \({\mathcal{P}^3({\bf V}^4,{\bf V}_4, {\bf R})}\) . In our work we will use this projective model of \({{\widetilde{\bf SL_{2}R}}}\) and in this manner the prisms and prism tilings can be visualized on the Euclidean screen of a computer.     

Hypercentralizing generalized skew derivations on left ideals in prime rings

Let $\mathcal{R }$ be a prime ring of characteristic different from $2, \mathcal{Q }_r$ the right Martindale quotient ring of $\mathcal{R }, \mathcal{C }$ the extended centroid of $\mathcal{R }, \mathcal{I }$ a nonzero left ideal of $\mathcal{R }, F$ a nonzero generalized skew derivation of $\mathcal{R }$ with associated automorphism $\alpha $ , and $n,k \ge 1$ be fixed integers. If $[F(r^n),r^n]_k=0$ for all $r \in \mathcal{I }$ , then there exists $\lambda \in \mathcal{C }$ such that $F(x)=\lambda x$ , for all $x\in \mathcal{I }$ . More precisely one of the following holds: (1) $\alpha $ is an $X$ -inner automorphism of $\mathcal{R }$ and there exist $b,c \in \mathcal{Q }_r$ and $q$ invertible element of $\mathcal{Q }_r$ , such that $F(x)=bx-qxq^{-1}c$ , for all $x\in \mathcal{Q }_r$ . Moreover there exists $\gamma \in \mathcal{C }$ such that $\mathcal{I }(q^{-1}c-\gamma )=(0)$ and $b-\gamma q \in \mathcal{C }$ ; (2) $\alpha $ is an $X$ -outer automorphism of $\mathcal{R }$ and there exist $c \in \mathcal{Q }_r, \lambda \in \mathcal{C }$ , such that $F(x)=\lambda x-\alpha (x)c$ , for all $x\in \mathcal{Q }_r$ , with $\alpha (\mathcal{I })c=0$ .     

A metrical lower bound on the star discrepancy of digital sequences

In this paper we study uniform distribution properties of digital sequences over a finite field of prime order. In 1998 it was shown by Larcher that for almost all $s$ -dimensional digital sequences $\mathcal{S }$ the star discrepancy $D_N^*$ satisfies an upper bound of the form $D_N^*(\mathcal{S })=O((\log N)^s (\log \log N)^{2+\varepsilon })$ for any $\varepsilon >0$ . Generally speaking it is much more difficult to obtain good lower bounds for specific sequences than upper bounds. Here we show that Larchers result is best possible up to some $\log \log N$ term. More detailed, we prove that for almost all $s$ -dimensional digital sequences $\mathcal{S }$ the star discrepancy satisfies $D_N^*(\mathcal{S }) \ge c(q,s) (\log N)^s \log \log N$ for infinitely many $N \in \mathbb{N }$ , where $c(q,s)>0$ only depends on $q$ and $s$ but not on $N$ .     

f-Vectors Implying Vertex Decomposability

We prove that if a pure simplicial complex $\Delta $ of dimension $d$ with $n$ facets has the least possible number of $(d-1)$ -dimensional faces among all complexes with $n$ faces of dimension $d$ , then it is vertex decomposable. This answers a question of J. Herzog and T. Hibi. In fact, we prove a generalization of their theorem using combinatorial methods.     

Algebraic Boundaries of \mathrm{SO}(2)-Orbitopes

Let $X\subset \mathbb{A }^{2r}$ X ? A 2 r be a real curve embedded into an even-dimensional affine space. We characterise when the $r$ r th secant variety to $X$ X is an irreducible component of the algebraic boundary of the convex hull of the real points $X(\mathbb{R })$ X ( R ) of $X$ X . This fact is then applied to $4$ 4 -dimensional $\mathrm{SO}(2)$ SO ( 2 ) -orbitopes and to the so called Barvinok–Novik orbitopes to study when they are basic closed semi-algebraic sets. In the case of $4$ 4 -dimensional $\mathrm{SO}(2)$ SO ( 2 ) -orbitopes, we find all irreducible components of their algebraic boundary.     

Diophantine approximation with sign constraints

Let $\alpha $ and $\beta $ be real numbers such that $1$ , $\alpha $ and $\beta $ are linearly independent over $\mathbb {Q}$ . A classical result of Dirichlet asserts that there are infinitely many triples of integers $(x_0,x_1,x_2)$ such that $|x_0+\alpha x_1+\beta x_2| < \max \{|x_1|,|x_2|\}^{-2}$ . In 1976, Schmidt asked what can be said under the restriction that $x_1$ and $x_2$ be positive. Upon denoting by $\gamma \cong 1.618$ the golden ratio, he proved that there are triples $(x_0,x_1,x_2) \in \mathbb {Z}^3$ with $x_1,x_2>0$ for which the product $|x_0 + \alpha x_1 + \beta x_2| \max \{|x_1|,|x_2|\}^\gamma $ is arbitrarily small. Although Schmidt later conjectured that $\gamma $ can be replaced by any number smaller than $2$ , Moshchevitin proved very recently that it cannot be replaced by a number larger than $1.947$ . In this paper, we present a construction of points $(1,\alpha ,\beta )$ showing that the result of Schmidt is in fact optimal. These points also possess strong additional Diophantine properties that are described in the paper.     

Stability results for the N-dimensional Schiffer conjecture via a perturbation method

Given a eigenvalue $\mu _{0m}^2$ of $-\Delta $ in the unit ball $B_1$ , with Neumann boundary conditions, we prove that there exists a class $\mathcal{D}$ of $C^{0,1}$ -domains, depending on $\mu _{0m} $ , such that if $u$ is a no trivial solution to the following problem $ \Delta u+\mu u=0$ in $\Omega , u=0$ on $\partial \Omega $ , and $ \int \nolimits _{\partial \Omega }\partial _{\mathbf{n}}u=0$ , with $\Omega \in \mathcal{D}$ , and $\mu =\mu _{0m}^2+o(1)$ , then $\Omega $ is a ball. Here $\mu $ is a eigenvalue of $-\Delta $ in $\Omega $ , with Neumann boundary conditions.     

Explicit Constructions of Centrally Symmetric k-Neighborly Polytopes and Large Strictly Antipodal Sets

We present explicit constructions of centrally symmetric $2$ -neighborly $d$ -dimensional polytopes with about $3^{d/2}\approx (1.73)^d$ vertices and of centrally symmetric $k$ -neighborly $d$ -polytopes with about $2^{{3d}/{20k^2 2^k}}$ vertices. Using this result, we construct for a fixed $k\ge 2$ and arbitrarily large $d$ and $N$ , a centrally symmetric $d$ -polytope with $N$ vertices that has at least $\left( 1-k^2\cdot (\gamma _k)^d\right) \genfrac(){0.0pt}{}{N}{k}$ faces of dimension $k-1$ , where $\gamma _2=1/\sqrt{3}\approx 0.58$ and $\gamma _k = 2^{-3/{20k^2 2^k}}$ for $k\ge 3$ . Another application is a construction of a set of $3^{\lfloor d/2 -1\rfloor }-1$ points in $\mathbb R ^d$ every two of which are strictly antipodal as well as a construction of an $n$ -point set (for an arbitrarily large $n$ ) in $\mathbb R ^d$ with many pairs of strictly antipodal points. The two latter results significantly improve the previous bounds by Talata, and Makai and Martini, respectively.