Recent zbMATH articles in MSC 52Chttps://www.zbmath.org/atom/cc/52C2022-01-14T13:23:02.489162ZWerkzeugDelone sets generated by square rootshttps://www.zbmath.org/1475.111382022-01-14T13:23:02.489162Z"Marklof, Jens"https://www.zbmath.org/authors/?q=ai:marklof.jensConsider an infinite sequence of real numbers $(\xi_{n})_{n\in N^*}$ and the patterns generated by the complex numbers
\[
z_{n}=\sqrt ne^{2\pi i\xi_{n}}.
\]
Before discussing the case $\xi_{n}=\alpha\sqrt n$, author explores the relationship between the distribution of the sequence $(\xi_{n})_{n} \bmod 1$ and the distribution of $(z_{n})_{n}$ in the complex plane $C$.
Definition 1. We say a point set is uniformly discrete if any two points are separated by a given minimum distance, and relatively dense if there is a given radius so that every ball of that radius contains at least one point.
Definition 2. A Delone set is a point set that satisfies both of these properties of definition 1.
In this work, author shows that the point set given by the values $\sqrt ne^{2\pi i\alpha\sqrt n}$ with $n=1,2,3,\ldots$ is a Delone set in the complex plane, for any $\alpha>0$. This complements \textit{S. Akiyama}'s recent work [``Spiral Delone sets and three distance theorem'', Nonlinearity 33, No. 5, 2533--2540 (2020; \url{doi:10.1088/1361-6544/ab74ad})] that $\sqrt ne^{2\pi i\alpha n}$ with $n=1,2,3,\ldots$ is a Delone set, if and only if $\alpha$ is badly approximated by rationals. A key difference is that Marklof's setting does not require Diophantine conditions on α. More precisely, we have using the classical definitions of the uniform distribution and the concept of asymptotic density [\textit{L. Kuipers} and \textit{H. Niederreiter}, Uniform distribution of sequences. New York etc.: John Wiley \& Sons (1974; Zbl 0281.10001)], the author has proved the following beautiful results.
Result 1. Let $(\xi_{n})_{n\in N^*}$ be a sequence of real numbers and let $(z_{n})_{n}$ be the corresponding sequence given by $z_{n}=\sqrt ne^{2\pi i\xi_{n}}$. Then $(z_{n})_{n}$ has asymptotic density $\rho=\pi^{-1}$ if and only if $(\xi_{n})_{n}$ is uniformly distributed $\bmod 1$.
Result 2. Let $(\xi_{n})_{n\in N^*}$ be a sequence of real numbers and let $(z_{n})_{n}$ be the corresponding sequence given by $z_{n}=\sqrt ne^{2\pi i\xi_{n}}$, and $h>0$, $R>0$, we denote by $g_{R}^{h}$ the minimal gap mod 1 between the fractional parts of the elements $\xi_{n}$. Then (a) $(z_{n})_{n}$ is uniformly discrete if and only if there exists $h>0$ such that $\inf Rg_{R}^{h}>0$; (b) $(z_{n})_{n}$ is relatively dense if and only if there exists h>0 such that $\sup Rg_{R}^{h}<\infty$.
Result 3. Let $\alpha>0$ and $\xi_{n}=\alpha\sqrt n$. Then $(z_{n})_{n}$ is a Delone set with asymptotic density $\pi^{-1}$.
Reviewer: Noureddine Daili (Sétif)Real and complex supersolvable line arrangements in the projective planehttps://www.zbmath.org/1475.141072022-01-14T13:23:02.489162Z"Hanumanthu, Krishna"https://www.zbmath.org/authors/?q=ai:hanumanthu.krishna"Harbourne, Brian"https://www.zbmath.org/authors/?q=ai:harbourne.brianIn the paper under review the authors study some combinatorial problems related to the geometry of supersolvable arrangements defined over the complex and real numbers. Let \(\mathcal{L} \subset \mathbb{P}^{2}_{\mathbb{C}}\) be an arrangement of \(s\) lines in the projective plane. A point \(p\) is called a modular point for \(\mathcal{L}\) if it is an intersection point with the additional property that whenever \(q\) is another crossing point, then the line through \(p\) and \(q\) is a configurational line of \(\mathcal{L}\). We say that \(\mathcal{L}\) is supersolvable if it has a modular point. The main aim of the paper under review is to provide a classification of complex supersolvable line arrangements with respect to the number of modular points. The first result of the paper tells us that if \(\mathcal{L}\) is a complex arrangement of \(s\) lines in the projective which is non-trivial (i.e. is not a pencil of lines), then it cannot have more than \(4\) modular points. Based on that observation, the authors provide a complete description of complex supersolvable line arrangements with \(3\) and \(4\) modular points. The ultimate goal of the paper under review was to verify a conjecture saying that if \(\mathcal{C}\) is a non-trivial complex supersolvable arrangement of \(s\) lines, then the number of double points \(t_{2}(\mathcal{L})\) is bounded from below by \(s/2\). It turned out that this conjecture is true, as it has been recently shown by \textit{T. Abe} in [``Double points of free projective line arrangements'', Int. Math. Res. Notices 145 (2020; \url{doi:10.1093/imrn/rnaa145})].
Here we sum up the above loose discussion by a concrete theorem. Given a supersolvable line arrangement \(\mathcal{L}\), if it has two or more modular points and they do no all have the same multiplicity, we say that \(\mathcal{L}\) is not homogeneous, but if all modular points of \(\mathcal{L}\) have the same multiplicity, we say that \(\mathcal{L}\) is homogeneous, and \(m\)-homogeneous if the common multiplicity is \(m\).
Theorem. Let \(\mathcal{L}\) be a line arrangement having \(\mu_{\mathcal{L}} > 0\) modular points over any field \(k\).
a) If \(\mathcal{L}\) is not homogeneous, then either \(\mathcal{L}\) is a near pencil or \(\mu_{\mathcal{L}} = 2\); if \(\mu_{\mathcal{L}} = 2\), then \(\mathcal{L}\) consists of \(a\geq 2\) lines through one modular point, \(b>a\) lines through the other modular point, and we have \(s = a+b-1\) lines and \((a-1)(b-1)\) double intersection points.
b) If \(\mathcal{L}\) has a modular point of multiplicity \(2\), then \(\mathcal{L}\) is a pencil of lines.
c) If \(\mathcal{L}\) is complex and homogeneous with \(m > 2\), then \(1 \leq \mu_{\mathcal{L}} \leq 4\). If \(\mu_{\mathcal{L}} \in \{3,4\}\), we have the following possibilities:
i) If \(\mu_{\mathcal{L}} = 4\), then \(s=6\), \(m=3\), the number of double points is \(3\), the number of triple points is \(4\).
ii) If \(\mu_{\mathcal{L}} = 3\), then \(m>3\) and, up to change of coordinates, \(\mathcal{L}\) consists of the lines defined by the linear factors of \[xyz(x^{m-2} - y^{m-2})(x^{m-2} - z^{m-2})(y^{m-2}-z^{m-2}),\] and hence \(s = 3m-3\), the number of double points is \(3m-6\), the number of triple points is \((m-2)^2\), and the number of \(m\)-fold points is equal to \(3\).
Reviewer: Piotr Pokora (Kraków)Crystallographic groups, strictly tessellating polytopes, and analytic eigenfunctionshttps://www.zbmath.org/1475.200852022-01-14T13:23:02.489162Z"Rowlett, Julie"https://www.zbmath.org/authors/?q=ai:rowlett.julie"Blom, Max"https://www.zbmath.org/authors/?q=ai:blom.max"Nordell, Henrik"https://www.zbmath.org/authors/?q=ai:nordell.henrik"Thim, Oliver"https://www.zbmath.org/authors/?q=ai:thim.oliver"Vahnberg, Jack"https://www.zbmath.org/authors/?q=ai:vahnberg.jackThe authors generalize the results of \textit{P. H. Bérard} [Invent. Math. 58, 179--199 (1980; Zbl 0434.35068)] and \textit{B. J. McCartin} [Appl. Math. Sci., Ruse 2, No. 57--60, 2891--2901 (2008; Zbl 1187.35144)] to all dimensions. They prove that the following are equivalent: the first Dirichlet eigenfunction for the Laplace eigenvalue equation on a polytope is real analytic, the polytope strictly tessellates space, and the polytope is the fundamental domain of a crystallographic Coxeter group. They also show that under any of these equivalent conditions, all of the eigenfunctions are trigonometric functions. They connect these topics to the Fuglede and Goldbach conjectures and give a purely geometric formulation of Goldbach's conjecture.
Reviewer: Erich W. Ellers (Toronto)On Gilp's group-theoretic approach to Falconer's distance problemhttps://www.zbmath.org/1475.280112022-01-14T13:23:02.489162Z"Yu, Han"https://www.zbmath.org/authors/?q=ai:yu.hanSummary: In this paper, we follow and extend a group-theoretic method introduced by Greenleaf-Iosevich-Liu-Palsson (GILP) to study finite points configurations spanned by Borel sets in \(\mathbb{R}^n\), \(n\geq 2\), \(n\in\mathbb{N}\). We remove a technical continuity condition in a GILP's theorem in [\textit{A. Greenleaf} et al., Rev. Mat. Iberoam. 31, No. 3, 799--810 (2015; Zbl 1329.52015)]. This allows us to extend the Wolff-Erdogan dimension bound for distance sets to finite points configurations with \(k\) points for \(k\in\{2,\dots,n+1\}\) forming a \((k-1)\)-simplex.A characterization of high order freeness for product arrangements and answers to Holm's questionshttps://www.zbmath.org/1475.320162022-01-14T13:23:02.489162Z"Abe, Takuro"https://www.zbmath.org/authors/?q=ai:abe.takuro"Nakashima, Norihiro"https://www.zbmath.org/authors/?q=ai:nakashima.norihiroIn the paper under review, the authors study the so-called \(m\)-freeness of hyperplane arrangements. Let \(\mathbb{K}\) be a field of characteristic zero and let \(V\) be an \(l\)-dimensional vector space over \(\mathbb{K}\). A central hyperplane arrangement \(\mathcal{A}\) is a finite collection of hyperplanes in \(V\) such that each contains the origin. We call \(\mathcal{A}\) an \(l\)-arrangement when we emphasize the dimension of \(V\). Denote by \(Q \in S:= \mathbb{K}[x_{1},\dots, x_{l}]\) a defining polynomial of \(\mathcal{A}\). For \(\mathbf{a} = (a_{1},\dots, a_{l})\in \mathbb{N}^{l}\) one uses multi-index notations, namely
\[
|\mathbf{a}|= a_{1} + \dots + a_{l}, \quad \mathbf{a}! = a_{1}! \cdots a_{l}!, \quad \text{and} \quad \partial^{\mathbf{a}} = \partial_{1}^{a_{1}} \cdots \partial_{l}^{a_{l}},
\]
where \(\partial_{i} := \frac{\partial}{\partial \, x_{i}}\). Define
\[
D^{(m)}(S) = \bigoplus_{|\mathbf{a}| = m}S \cdot \partial^{\mathbf{a}}
\]
for \(m\geq 1\) and \(D^{(0)}(S) = S\). One can observe that \(D^{(m)}(S)\) is an \(S\)-submodule of \(\mathrm{End}_{\mathbb{K}}(S)\). For a central hyperplane arrangement \(\mathcal{A}\) we define an \(S\)-submodule \(D^{(m)}(\mathcal{A})\) of \(D^{(m)}(S)\) by
\[
D^{(m)}(\mathcal{A}) = \{ \theta \in D^{(m)}(S) \, | \, \theta(QS) \subset QS\}.
\]
We call \(D^{(m)}(\mathcal{A})\) the module of \(m\)-th order \(\mathcal{A}\)-differential operators. Finally, we say that \(\mathcal{A}\) is \(m\)-free if \(D^{(m)}(\mathcal{A})\) is a free \(S\)-module.
Since \(QD^{(m)}(S) \subseteq D^{(m)}(\mathcal{A}) \subset D^{(m)}(S)\), the rank of \(D^{(m)}(\mathcal{A})\) is \(s = s_{m}(l) = \binom{l+m-1}{m}\) provided that \(\mathcal{A}\) is \(m\)-free.
We say that \(\theta = \sum_{|\mathbf{a}|=m}f_{\mathbf{a}}\partial^{\mathbf{a}} \in D^{(m)}(S)\) is homogeneous of degree \(j\) and write \(\deg (\theta) = j\) if \(f_{\mathbf{a}}\) is zero or homogeneous of degree \(j\) for each \(\mathbf{a}\). If \(m\geq 1\) and if \(\mathcal{A}\) is \(m\)-free with a homogeneous basis \(\{\theta_{1},\dots, \theta_{s}\}\), we define \(m\)-exponents by the multi-set \(\exp_{m}(\mathcal{A}) = \{ \deg (\theta_{1}), \dots, \deg (\theta_{s})\}\).
In the context of further developments on the \(m\)-freeness, one can ask the following questions (proposed by P. Holm around 2002):
Q1) Does \(m\)-free imply \((m+1)\)-free for any arrangement?
Q2) Are all arrangements \(m\)-free for \(m\) large enough?
It turns out, however, that the answers to the above two questions are negative, and this is the main core of the present paper under review.
Theorem A. Let \((\mathcal{A}_{1}, V_{1})\) and \((\mathcal{A}_{2}, V_{2})\) be arrangements with \(\dim V_{1} > 0\) and \(\dim V_{2} > 0\). The following conditions are equivalent:
a) \((\mathcal{A}_{1} \times \mathcal{A}_{2}, V_{1} \oplus V_{2})\) is \(m\)-free.
b) Both \((\mathcal{A}_{1}, V_{1})\) and \((\mathcal{A}_{2}, V_{2})\) are \(i\)-free for all \(1 \leq i \leq m\).
c) \((\mathcal{A}_{1} \times \mathcal{A}_{2}, V_{1} \oplus V_{2})\) is \(i\)-free for all \(1\leq i \leq m\).
This result implies that the answer to Q2 is no.
Example. Let \(\mathcal{A}\) and \(\mathcal{A}'\) be two arrangements. If \(\mathcal{A}\) is not \(1\)-free, then the product arrangement \(\mathcal{A} \times \mathcal{A}'\) is not \(m\)-free for any \(m\geq 1\). In particular, a generic arrangement is known to be not \(1\)-free and hence if \(\mathcal{A}\) is generic and \(\mathcal{A}'\) is arbitrary, then \(\mathcal{A} \times \mathcal{A}'\) is not \(m\)-free for any \(m\geq 1\).
For \(l\geq 2\) we define \(\mathrm{Shi}_{l}\) as an \((l+1)\)-arrangement defined by
\[
Q(\mathrm{Shi}_{l}) = z \prod_{i=1}^{l}x_{i}(x_{i}-z) \prod_{1\leq i < j\leq l}(x_{i}-x_{j})(x_{i}-x_{j}-z).
\]
Theorem B. The arrangement \(\mathrm{Shi}_{l}\) is not \(2\)-free for all \(l\geq 2\).
Since \(\mathrm{Shi}_{l}\) is known to be \(1\)-free thus the answer to Q1 is also no.
Reviewer: Piotr Pokora (Kraków)Platonic solids, Archimedean solids and semi-equivelar maps on the spherehttps://www.zbmath.org/1475.520142022-01-14T13:23:02.489162Z"Datta, Basudeb"https://www.zbmath.org/authors/?q=ai:datta.basudeb"Maity, Dipendu"https://www.zbmath.org/authors/?q=ai:maity.dipenduSummary: A map \(X\) on a surface is called vertex-transitive if the automorphism group of \(X\) acts transitively on the set of vertices of \(X\). A map is called semi-equivelar if the cyclic arrangement of faces around each vertex is same. In general, semi-equivelar maps on a surface form a bigger class than vertex-transitive maps. There are semi-equivelar maps on the torus, the Klein bottle and other surfaces which are not vertex-transitive. It is known that the boundaries of Platonic solids, Archimedean solids, regular prisms and anti-prisms are vertex-transitive maps on \(\mathbb{S}^2\). Here we show that there is exactly one semi-equivelar map on \(\mathbb{S}^2\) which is not vertex-transitive. As a consequence, we show that all the semi-equivelar maps on \(\mathbb{RP}^2\) are vertex-transitive. Moreover, every semi-equivelar map on \(\mathbb{S}^2\) can be geometrized, i.e., every semi-equivelar map on \(\mathbb{S}^2\) is isomorphic to a semi-regular tiling of \(\mathbb{S}^2\). In the course of the proof of our main result, we present a combinatorial characterisation in terms of an inequality of all the types of semi-equivelar maps on \(\mathbb{S}^2\). Here we present combinatorial proofs of all the results.Specified holes with pairwise disjoint interiors in planar point setshttps://www.zbmath.org/1475.520272022-01-14T13:23:02.489162Z"Hosono, Kiyoshi"https://www.zbmath.org/authors/?q=ai:hosono.kiyoshi"Urabe, Masatsugu"https://www.zbmath.org/authors/?q=ai:urabe.masatsuguSome \(n\) points in the plane are said to be in general position if no three of them are collinear. A well-known conjecture of \textit{P. Erdős} and \textit{G. Szekeres} [Compos. Math. 2, 463--470 (1935; Zbl 0012.27010)] asserts that among \(2^{n-2}+1\) points in general position in the plane, some \(n\) points are in convex position. A recent breakthrough result of \textit{A. Suk} [J. Am. Math. Soc. 30, No. 4, 1047--1053 (2017; Zbl 1370.52032)] came close to proving this conjecture by showing that among \(2^{n+O(n^{2/3}\log n)}\) points in general position in the plane, some \(n\) points are in convex position.
Erdős also posed the following problem: Find the smallest integer \(n(k)\), such that any set of \(n(k)\) points in general position in the plane, contains the vertices of a convex \(k\)-gon, whose interior contains no points of the set. Such a subset is called a \(k\)-hole of the set. \textit{J. D. Horton} [Can. Math. Bull. 26, 482--484 (1983; Zbl 0521.52010)] constructed arbitrarily large point sets which do not contain any 7-holes, so \(n(k)\) does not exist for \(k \geq 7\).
A pair of holes is said to be disjoint if their convex hulls do not intersect.
For \(k\leq \ell\), let \( n(k, \ell)\) denote the smallest integer such that any set of \(n(k, \ell)\) points in general position in the plane, contains both a \(k\)-hole and an \(\ell\)-hole, which are disjoint.
A family of holes is with disjoint interiors if their interiors are pairwise disjoint. For \(k\leq \ell\), let \(m(k, \ell)\) denote the smallest integer such that any set of \(m(k, \ell)\) points in general position in the plane contains both a \(k\)-hole and an \(l\)-hole with disjoint interiors. Obviously \(m(k, \ell) \leq n(k, \ell)\) holds; and \( m(k, \ell)\) does not exist for any \(\ell\geq 7\) by Horton's cited result. The paper under review studies the \(m(k, \ell)\) numbers, and their analogues for more than two holes.
Reviewer: László A. Székely (Columbia)Upper and lower bounds for rich lines in gridshttps://www.zbmath.org/1475.520282022-01-14T13:23:02.489162Z"Murphy, Brendan"https://www.zbmath.org/authors/?q=ai:murphy.brendanSummary: We prove upper and lower bounds for the number of lines in general position that are rich in a Cartesian product point set. This disproves a conjecture of Solymosi and improves work of \textit{G. Elekes} [Combinatorica 18, No. 1, 13--25 (1998; Zbl 0923.11029); Bolyai Soc. Math. Stud. 11, 241--290 (2002; Zbl 1060.11013)], \textit{E. Borenstein} and \textit{E. Croot} [Discrete Comput. Geom. 43, No. 4, 824--840 (2010; Zbl 1192.52026)], and \textit{G. Amirkhanyan} et al. [J. Lond. Math. Soc., II. Ser. 96, No. 1, 67--85 (2017; Zbl 1375.52016)].
The upper bounds are based on a version of the asymmetric Balog-Szemerédi-Gowers theorem for group actions combined with product theorems for the affine group. The lower bounds are based on a connection between rich lines in Cartesian product sets and amenability (or expanding families of graphs in the finite field case).
As an application of our upper bounds for rich lines in grids, we give a geometric proof of the asymmetric sum-product estimates of \textit{J. Bourgain} [Int. J. Number Theory 1, No. 1, 1--32 (2005; Zbl 1173.11310)] and \textit{I. Shkredov} [Mosc. J. Comb. Number Theory 8, No. 1, 15--41 (2019; Zbl 1454.11029)].On the number of discrete chainshttps://www.zbmath.org/1475.520292022-01-14T13:23:02.489162Z"Palsson, Eyvindur Ari"https://www.zbmath.org/authors/?q=ai:palsson.eyvindur-ari"Senger, Steven"https://www.zbmath.org/authors/?q=ai:senger.steven"Sheffer, Adam"https://www.zbmath.org/authors/?q=ai:sheffer.adamThe paper under review generalizes the still wide open Erdős unit distance problem. For \(n\) points in the Euclidean plane, a fixed number \(k\), and distances \(\delta_1,\delta_2,\ldots,\delta_k\), how large can be the number of \(p_1,p_2,\ldots,p_{k+1}\) sequences selected from the \(n\) points, such that for every \(i\), the distance between \(p_{i}\) and \(p_{i+1}\) is \(\delta_i\)? The Erdős unit distance problem corresponds to the \(k=1\) instance of this problem. The authors provide constructions for lower bounds and proofs for upper bounds for this quantity, and conjecture that the lower bounds are about tight.
I do not describe the upper bounds of the paper in detail, as those were superseded by \textit{N. Frankl} and \textit{A. Kupavskii} [``Almost sharp bounds on the number of discrete chains in the plane'', Preprint, \url{arXiv:1912.00224}]. Their arXiv report determined the correct growth of the function of \(n\) for the planar problem, when \(k\equiv 0\) or \(2\) mod \(3\), and almost determined it when \(k\equiv 1\) mod \(3\), in which case the answer involves the solution to the Erdős unit distance problem. These results essentially confirmed the conjectures of the paper under review. There are results for the analogous problem in the 3-dimensional Euclidean space as well, on which Frankl and Kupavskii [loc. cit.] also made improvement.
Reviewer: László A. Székely (Columbia)Brick partition problems in three dimensionshttps://www.zbmath.org/1475.520302022-01-14T13:23:02.489162Z"Choi, Ilkyoo"https://www.zbmath.org/authors/?q=ai:choi.ilkyoo"Kim, Minseong"https://www.zbmath.org/authors/?q=ai:kim.minseong"Seo, Kiwon"https://www.zbmath.org/authors/?q=ai:seo.kiwonA (3D) brick is the Cartesian product of three closed intervals. A brick partition is a brick cut into brick pieces; it is called \(k\)-piercing (resp. \(k\)-slicing) if every cutting axis-parallel line (resp. plane) cuts at least \(k\) of its pieces.
The minimum number of pieces of a 2-piercing brick partition is 8, and for 2-slicing it is 4.
For \(k\geq 3\), a \(k\)-piercing brick partition with \(12k-15\) pieces is constructed, while any such partition is shown to need at least \(12k-16\) pieces. The minimum number of pieces of a \(k\)-slicing brick partition is shown to be exactly \(2k-1\).
Reviewer: Frank Plastria (Brussels)New proofs of infinitesimal rigidity of convex polyhedra and convex surfaces of revolutionhttps://www.zbmath.org/1475.520312022-01-14T13:23:02.489162Z"Sabitov, Idzhad Khakovich"https://www.zbmath.org/authors/?q=ai:sabitov.idzhad-khNew proofs are presented for the following two theorems, relevant to the properties of infinitesimal rigidity of convex polyhedra and convex surfaces of revolution.
Theorem 1. Strictly convex polyhedra with rigid faces are rigid with respect to infinitesimal hoggings of the first order.
Theorem 2. Closed convex surface of revolution without flat parts is rigid.
Reviewer: Boris Ivanovich Konosevich (Donetsk)Characteristic polynomials of linial arrangements for exceptional root systemshttps://www.zbmath.org/1475.520322022-01-14T13:23:02.489162Z"Yoshinaga, Masahiko"https://www.zbmath.org/authors/?q=ai:yoshinaga.masahikoA hyperplane arrangement is a finite collection of affine hyperplanes in a vector space. The characteristic polynomial of a hyperplane is an important invariant. A linial arrangement is defined via \[\mathcal{L}^m_{\Phi} =\{H_{\alpha, k} \mid \alpha\in \Phi^+, k=1, \ldots m\}\] where \(\Phi\) is an irreducible root system. Combinatorial properties of linial arrangements were studied by \textit{A. Postnikov} and \textit{R. P. Stanley} in [J. Comb. Theory, Ser. A 91, No. 1--2, 544--597 (2000; Zbl 0962.05004)], and in that paper, they conjectured that the roots of the characteristic polynomial of a linial arrangement all have real part equal to \(\frac{mh}{2}\) where \(h\) is the Coxeter number of \(\Phi\). With this current paper, the author proves this conjecture for exceptional root systems and \(m>>0\).
Reviewer: Kelly J. Pearson (Murray)On the number of faces and radii of cells induced by Gaussian spherical tessellationshttps://www.zbmath.org/1475.600302022-01-14T13:23:02.489162Z"Lybrand, Eric"https://www.zbmath.org/authors/?q=ai:lybrand.eric"Ma, Anna"https://www.zbmath.org/authors/?q=ai:ma.anna"Saab, Rayan"https://www.zbmath.org/authors/?q=ai:saab.rayanSummary: We study a geometric property related to spherical hyperplane tessellations in \(\mathbb{R}^d\). We first consider a fixed \(x\) on the Euclidean sphere and tessellations with \(M\gg d\) hyperplanes passing through the origin having normal vectors distributed according to a Gaussian distribution. We show that with high probability there exists a subset of the hyperplanes whose cardinality is on the order of \(d\log(d)\log(M)\) such that the radius of the cell containing \(x\) induced by these hyperplanes is bounded above by, up to constants, \(d\log(d)\log(M)/M\). We extend this result to hold for all cells in the tessellation with high probability. Up to logarithmic terms, this upper bound matches the previously established lower bound of \textit{V. K. Goyal} et al. [IEEE Trans. Inf. Theory 44, No. 1, 16--31 (1998; Zbl 0905.94007)].Tilings of \((2 \times 2 \times n)\)-board with coloured cubes and brickshttps://www.zbmath.org/1475.970432022-01-14T13:23:02.489162Z"Németh, László"https://www.zbmath.org/authors/?q=ai:nemeth.laszlo.1|nemeth.laszloSummary: Several articles deal with tilings with squares and dominoes on 2-dimensional boards, but only a few on boards in 3-dimensional space. We examine a tiling problem with coloured cubes and bricks on a \((2 \times 2 \times n)\)-board in three dimensions. After a short introduction and the definition of breakability we show a way to get the number of the tilings of an \(n\)-long board considering the \((n - 1)\)-long board. It describes recursively the number of possible breakable and unbreakable tilings. Finally, we give some identities for the recursions using breakability.