Magdalena Łysakowska, Krzysztof Przesławski
It is shown that if n<7, then each tiling of R^n by translates of the unit cube [0,1)^n contains a column; that is, a family of the form {[0,1)^n+(s+ke_i): k \in Z}, where s \in R^n, e_i is an element of the standard basis of R^n and Z is the set of integers.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
O. H. Keller conjectured in 1930 that in any tiling of $\Bbb R^n$ by unit $n$-cubes there exist two of them having a complete facet in common. O. Perron proved this conjecture for $n\le 6$. We show that for all $n\ge 10$ there exists a tiling of $\Bbb R^n$ by unit $n$-cubes such that no two $n$-cubes have a complete facet in common.
Keller properties for integer tiling
本文研究整数平铺中Keller猜想的类似物，该猜想涉及立方体平铺中共享面的存在性，已知在低维成立、高维失败。
A cube tiling of $\mathbb{R}^d$ is a family of pairwise disjoint cubes $[0,1)^d+T=\{[0,1)^d+t\colon t\in T\}$ such that $\bigcup_{t\in T}([0,1)^d+t)=\mathbb{R}^d$. Two cubes $[0,1)^d+t$, $[0,1)^d+s$ are called a twin pair if $|t_j-s_j|=1$ for some $j\in [d]=\{1,\ldots, d\}$ and $t_i=s_i$ for every $i\in [d]\setminus \{j\}$. In $1930$, Keller conjectured that in every cube tiling of $\mathbb{R}^d$ there is a twin pair. Keller's conjecture is true for dimensions $d\leq 6$ and false for all dimensions $d\geq 8$. For $d=7$ the conjecture is still open. Let $x\in \mathbb{R}^d$, $i\in [d]$, and let …
We consider sequential random packing of integral translate of cubes $[0,N]^n$ into the torus $Z^n / 2NZ^n$. Two special cases are of special interest: (i) The case $N=2$ which corresponds to a discrete case of tilings (considered in \cite{cubetiling,book}) (ii) The case $N=\infty$ corresponds to a case of continuous tilings (considered in \cite{combincubepack,book}) Both cases correspond to some special combinatorial structure and we describe here new developments.
A cube tiling of R^d is a family of pairwise disjoint cubes $[0,1)^d+T=\{[0,1)^d+t:t\in T\}$ such that $\bigcup_{t\in T}([0,1)^d+t)=R^d$. Two cubes $[0,1)^d+t$, $[0,1)^d+s$ are called a twin pair if their closures have a complete facet in common, that is if $|t_j-s_j|=1$ for some $j\in [d]=\{1,..., d\}$ and $t_i=s_i$ for every $i\in [d]\setminus \{j\}$. In 1930, Keller conjectured that in every cube tiling of R^d there is a twin pair. Keller's conjecture is true for dimensions $d\leq 6$ and false for all dimensions $d\geq 8$. For $d=7$ the conjecture is still open. Let $x\in R^d$, $i\in [d]$, …
Let $S$ be a set of arbitrary objects, and let $s\mapsto s'$ be a permutation of $S$ such that $s"=(s')'=s$ and $s'\neq s$. Let $S^d=\{v_1...v_d\colon v_i\in S\}$. Two words $v,w\in S^d$ are dichotomous if $v_i=w'_i$ for some $i\in [d]$, and they form a twin pair if $v_i'=w_i$ and $v_j=w_j$ for every $j\in [d]\setminus \{i\}$. A polybox code is a set $V\subset S^d$ in which every two words are dichotomous. A polybox code $V$ is a cube tiling code if $|V|=2^d$. A $2$-periodic cube tiling of $\mathbb{R}^d$ and a cube tiling of flat torus $\mathbb{T}^d$ can be encoded in a form of a cube tiling c…
Using a cube tiling of $\mathbb{R}^n$ constructed by Lagarias and Shor a tiling proof of three well-known binomial identities related to the Lucas cube is given.
A cube tiling of $\mathbb{R}^d$ is a family of pairwise disjoint cubes $[0,1)^d+T=\{[0,1)^d+t:t\in T\}$ such that $\bigcup_{t\in T}([0,1)^d+t)=\mathbb{R}^d$. Two cubes $[0,1)^d+t$, $[0,1)^d+s$ are called a twin pair if $|t_j-s_j|=1$ for some $j\in [d]=\{1,\ldots, d\}$ and $t_i=s_i$ for every $i\in [d]\setminus \{j\}$. In $1930$, Keller conjectured that in every cube tiling of $\mathbb{R}^d$ there is a twin pair. Keller's conjecture is true for dimensions $d\leq 6$ and false for all dimensions $d\geq 8$. For $d=7$ the conjecture is still open. Let $x\in \mathbb{R}^d$, $i\in [d]$, and let $L(T,x…
Let $S$ be a set of arbitrary objects, and let $S^d=\{v_1...v_d\colon v_i\in S\}$. A polybox code is a set $V\subset S^d$ with the property that for every two words $v,w\in V$ there is $i\in [d]$ with $v_i'=w_i$, where a permutation $s\mapsto s'$ of $S$ is such that $s''=(s')'=s$ and $s'\neq s$. If $|V|=2^d$, then $V$ is called a cube tiling code. Cube tiling codes determine $2$-periodic cube tilings of $\mathbb{R}^d$ or, equivalently, tilings of the flat torus $\mathbb{T}^d=\{(x_1,\ldots ,x_d)({\rm mod} 2):(x_1,\ldots ,x_d)\in \mathbb{R}^d\}$ by translates of the unit cube as well as $r$-perf…
A non-empty subset A of X = X_1 x...x X_d is a (proper) box if A = A_1 x...x A_d and A_i is a (proper) subset of X_i for each i. Suppose that for each pair of boxes A, B and each i, one can only know which of the three states takes place: A_i = B_i, A_i = X_i\setminus B_i, A_i is different from each of the sets B_i, X_i\setminus B_i. Let F and G be two systems of disjoint boxes. Can one decide whether their unions are equal? In general, the answer is no, but as is shown in the paper, it is yes if both systems consist of pairwise dichotomous boxes. Several criteria that enable to compare such s…
Extremal length is a conformal invariant that transfers naturally to the discrete setting, giving square tilings as a natural combinatorial analog of conformal mappings. Recent work by S. Hersonsky has explored generalizing these ideas to three-dimensional cube tilings. The connections between discrete extremal length and cube tilings survive the dimension jump, but a condition called the triple intersection property is needed to generalize existence arguments. We show that this condition is too strong to realize a tiling, thus showing that discrete conformal mappings are far more limited in d…
The Resolution of Keller's Conjecture
本文通过将7维Keller猜想中128规模团的存在性编码为命题公式，并结合SAT求解与对称性破缺，证明相关图中不存在该团，从而完全解决了Keller猜想。
Given a triangulation of a closed topological cube, we show that (under some technical condition) there is an essentially unique tiling of a rectangular parallelepiped by cubes, indexed by the vertices of the triangulation. Moreover, i - the combinatorics is preserved, and ii- the boundary is preserved: vertices corresponding to the cubes at the corners of the rectangular parallelepiped are at the corners of the topological cube. Also, the sizes of the cubes are obtained as a solution of a variational problem which is a discrete version of the notion of extremal length in three dimensional Euc…
We show that three dimensional cubes of any size can be tiled with trominoes and, when necessary, one or two singletons in any positions. Cubes of side length a multiple of three can always be tiled with trominoes (known), cubes of side length congruent to 1 mod 3 can always be tiled with an arbitrary single cube and trominoes, and cubes of side length congruent to 2 mod 3 can always be tiled with two single cubes in arbitrary locations and trominoes.
Keller packings and tilings of boxes are investigated. Certain general inequality measuring a complexity of such systems is proved. A straightforward application to the unit cube tilings is given.
We describe odd-length-cube tilings of the n-dimensional q-ary torus what includes q-periodic integer lattice tilings of R^n. In the language of coding theory these tilings correspond to perfect codes with respect to the maximum metric. A complete characterization of the two-dimensional tillings is presented and in the linear case, a description of general matrices, isometry and isomorphism classes is provided. Several methods to construct perfect codes from codes of smaller dimension or via sections are derived. We introduce a special type of matrices (perfect matrices) which are in correspon…
Let k_1,...,k_d be positive integers, and D be a subset of [k_1]x...x[k_d], whose complement can be decomposed into disjoint sets of the form {x_1}x...x{x_{s-1}}x[k_s]x{x_{s+1}}x...x{x_d}. We conjecture that the number of elements of D can be represented as a linear combination of the numbers k_1,..., k_d with non-negative integer coefficients. A connexion of this conjecture with the structure of periodical cube tilings is revealed.
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 colored cubes and bricks of $(2\times2\times n)$-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. The method of determining the recur…
Alternating links, rational balls, and cube tilings
针对沿交错链环分支的三维球面二重覆叠的有理同调球边界问题，提出棋盘格点立方遍在性充要猜想，证明归一化行列式为1的情形，并将结果推广至缎带配边与协和性研究。
We consider tilings and packings of $\RR^d$ by integral translates of cubes $[0,2[^d$, which are $4\ZZ^d$-periodic. Such cube packings can be described by cliques of an associated graph, which allow us to classify them in dimension $d\leq 4$. For higher dimension, we use random methods for generating some examples. Such a cube packing is called {\em non-extendible} if we cannot insert a cube in the complement of the packing. In dimension 3, there is a unique non-extendible cube packing with 4 cubes. We prove that $d$-dimensional cube packings with more than $2^d-3$ cubes can be extended to cub…
A tiling of the $n$-dimensional Hamming cube gives rise to a perfect code (according to a given metric) if the basic tile is a metric ball. We are concerned with metrics on the $n$-dimensional Hamming cube which are determined by a weight which respects support of vectors (TS-metrics). We consider the known tilings of the Hamming cube and first determine which of them give rise to a perfect code. In the sequence, for those tilings that satisfy this condition, we determine all the TS-metrics that turns it into a perfect code. We also propose the construction of new perfect codes obtained by the…
We introduce a recurrence which we term the multidimensional cube recurrence, generalizing the octahedron recurrence studied by Propp, Fomin and Zelevinsky, Speyer, and Fock and Goncharov and the three-dimensional cube recurrence studied by Fomin and Zelevinsky, and Carroll and Speyer. The states of this recurrence are indexed by tilings of a polygon with rhombi, and the variables in the recurrence are indexed by vertices of these tilings. We travel from one state of the recurrence to another by performing elementary flips. We show that the values of the recurrence are independent of the order…
In this paper we introduce a new algebraic method in tilings. Combining this method with Hilbert's Nullstellensatz we obtain a necessary condition for tiling $n$-space by translates of a cluster of cubes. Further, the polynomial method will enable us to show that if there exists a tiling of $n$-space by translates of a cluster $V$ of prime size then there is a lattice tiling by $V$ as well. Finally, we provide supporting evidence for a conjecture that each tiling by translates of a prime size cluster $V$ is lattice if $V$ generates $n$-space.
An $n$-dimensional cross comprises $2n+1$ unit cubes: the center cube and reflections in all its faces. It is well known that there is a tiling of $R^{n}$ by crosses for all $n.$ AlBdaiwi and the first author proved that if $2n+1$ is not a prime then there are $2^{\aleph_{0}}$ \ non-congruent regular (= face-to-face) tilings of $R^{n}$ by crosses, while there is a unique tiling of $R^{n}$ by crosses for $n=2,3$. They conjectured that this is always the case if $2n+1$ is a prime. To support the conjecture we prove in this paper that also for $R^{5}$ there is a unique regular, and no non-regular…
Cube moves for $s$-embeddings and $α$-realizations
本文证明 $s$-嵌入中立方体移动的存在唯一性，给出共轭 Ising 星-三角变换的新耦合常数公式，并推广至 $α$-嵌入以统一调和嵌入与 $s$-嵌入。
The minimum surface area of $k$ unequal boxes tiling a cube: sharp thresholds, a fault-free law, and a reduction to two dimensions
研究用\(k\)个不等盒子平铺立方体的最小表面积问题，通过确定\(k = 6\)时的情况及规律，利用无故障定律、降维定理和加倍定律等方法，得到相关结果并给出特定区域无需计算机的证明。
The sharp diagonal spectral correlation inequality on the discrete cube
证明Friedgut-Kahn-Kalai-Keller提出的尖锐对角谱相关猜想，给出Harris-Kleitman不等式的傅里叶强化，并刻画所有等号情形。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。