Related papers: Hypercubic Self-Tilings
We consider the possible sizes of large sumfree sets contained in the discrete hypercube $\{1,...,n\}^k$, and we determine upper and lower bounds for the maximal size as $n$ becomes large. We also discuss a continuous analogue in which our…
We provide a list of conjectured surface-area-minimizing n-hedral tiles of space for n from 4 to 14, previously known only for n equal to 5 or 6. We find the optimal "orientation-preserving" tetrahedral tile (n=4), and we give a nice proof…
Consider the $n$-cube graph with vertices $\{-1,1\}^n$ and edges connecting vertices with hamming distance $1$. How many hyperplanes in $\mathbb{R}^n$ are needed in order to dissect all edges? We show that at least…
Twisted hypercubes are generalizations of the Boolean hypercube, obtained by iteratively connecting two instances of a graph by a uniformly random perfect matching. Dudek et al. showed that when the two instances are independent, these…
The ternary Goldbach conjecture states that every odd number n>=7 is the sum of three primes. The estimation of sums of the form \sum_{p\leq x} e(\alpha p), \alpha = a/q + O(1/q^2), has been a central part of the main approach to the…
The Thue-Siegel method is used to obtain an upper bound for the number of primitive integral solutions to a family of quartic Thue's inequalities. This will provide an upper bound for the number of integer points on a family of elliptic…
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…
We give a simple proof of T. Stehling's result, that in any normal tiling of the plane with convex polygons with number of sides not less than six, all tiles except the finite number are hexagons.
We investigate lower bounds for the number of ideal and finite vertices of right-angled hyperbolic polyhedra of finite volume. We use a geometric method of orthogonal gluings to establish new bounds in low dimensions, specifically…
Recently, Greenfeld and Tao disprove the conjecture that translational tilings of a single tile can always be periodic [Ann. Math. 200(2024), 301-363]. In another paper [to appear in J. Eur. Math. Soc.], they also show that if the dimension…
Given a graph G and a configuration C of pebbles on the vertices of G, a pebbling step removes two pebbles from one vertex and places one pebble on an adjacent vertex. The cover pebbling number g=g(G) is the minimum number so that every…
We introduce a family of nonnegative integer vectors - primitive vectors - defining hyperplanes of the real affine cube over $C^n:=\{-1,1\}^n$ and study their properties with respect to the rectangles of the cube. As a consequence we give a…
The Calisson puzzle is a tiling puzzle in which one must tile a triangular grid inside a hexagon with lozenges, under the constraint that certain prescribed edges remain tile boundaries and that adjacent lozenges along these edges have…
The $n$-th Fibonacci cube $\Gamma_n$ is the subgraph of the hypercube $Q_n$ induced by binary strings with no two consecutive ones. We determine $\pi(\Gamma_n) = 2^n$ for $n \le 6$, so the pebbling number of $\Gamma_n$ equals that of the…
An explicit upper bound is established for the least non-trivial integer zero of an arbitrary cubic form $C \in \mathbb{Z}[X_1,...,X_n],$ provided that $n \geq 14.$
We study the set of visible lattice points in multidimensional hypercubes. The problems we investigate mix together geometric, probabilistic and number theoretic tones. For example, we prove that almost all self-visible triangles with…
We consider the tiling of an $n$-board (a $1\times n$ array of square cells of unit width) with half-squares ($\frac12\times1$ tiles) and $(\frac12,\frac12)$-fence tiles. A $(\frac12,\frac12)$-fence tile is composed of two half-squares…
Let $ex(Q_n, H)$ be the largest number of edges in a subgraph $G$ of a hypercube $Q_n$ such that there is no subgraph of $G$ isomorphic to $H$. We show that for any integer $k\geq 3$, $$ex(Q_n, C_{4k+2})= O(n^{\frac{5}{6} +…
We show that the size of a minimal simplicial cover of a polytope $P$ is a lower bound for the size of a minimal triangulation of $P$, including ones with extra vertices. We then use this fact to study minimal triangulations of cubes, and…
Lebesgue's universal covering problem is re-examined using computational methods. This leads to conjectures about the nature of the solution which if correct could provide a blueprint for a complete solution. Empirical lower bounds for the…