Related papers: Long shortest vectors in low dimensional lattices

Suppose that $M$ is a finitely-generated graded module of codimension $c\geq 3$ over a polynomial ring and that the regularity of $M$ is at most $2a-2$ where $a\geq 2$ is the minimal degree of a first syzygy of $M$. Then we show that the…

This paper primarily studies monomial ideals by their associated lcm-lattices. It first introduces notions of weak coordinatizations of finite atomic lattices which have weaker hypotheses than coordinatizations and shows the…

A distributive lattice $L$ with minimum element $0$ is called decomposable lattice if $a$ and $b$ are not comparable elements in $L$ there exist $\overline{a},\overline{b}\in L$ such that $a=\overline{a}\vee(a\wedge b),…

We prove identities generating higher dimensional vector partitions. We derive theorems for integer lattice points in the 2D first quadrant, then generalize the approach to find 3D and $n$-space lattice point vector region extensions. We…

For a positive integer n, we denote by SUB (resp., SUBn) the class of all lattices that can be embedded into the lattice Co(P) of all order-convex subsets of a partially ordered set P (resp., P of length at most n). We prove the following…

We consider the set $A_{n}=\displaystyle\cup_{j=0}^{\infty}\{\varepsilon_{j}(n)\cdot n^j\colon\varepsilon_{j}(n)\in\{0,\pm1,\pm2,...,\pm\lfloor{{n}/{2}}\rfloor\}\} $. Let $\mathcal{S}_{\mathcal{A}}= \bigcup_{a \in\mathcal{A} } A_{a}$ where…

In this paper, we establish the explicit lower bound estimates for the rank of universal quadratic forms in some certain families of real cubic fields under the condition of density one. The more general results that represent all multiples…

The Systematic Normal Form (SysNF) is a canonical form of lattices introduced in [Eldar,Shor '16], in which the basis entries satisfy a certain co-primality condition. Using a "smooth" analysis of lattices by SysNF lattices we design a…

We explain why naive discretization results that have appeared in [hep-lat/0006013] do not appear to yield the desired continuum limit. The fermion propagator on the lattice inevitably yields a diagram with nonvanishing UV degree D=0…

The sets of all neutral, distributive and lower-modular elements of the lattice of semigroup varieties are finite, countably infinite and uncountably infinite, respectively. In 2018, we established that there are precisely three neutral…

In this paper we answer a question of Gabriel Navarro about orbit sizes of a finite linear group H acting completely reducibly on a vector space V: if the orbits containing the vectors a and b have coprime lengths m and n, we prove that the…

$ $Abert, Gelander and Nikolov [AGN17] conjectured that the number of generators $d(\Gamma)$ of a lattice $\Gamma$ in a high rank simple Lie group $H$ grows sub-linearly with $v = \mu(H / \Gamma)$, the co-volume of $\Gamma$ in $H$. We prove…

We investigate minimal-perimeter configurations of two finite sets of points on the square lattice. This corresponds to a lattice version of the classical double-bubble problem. We give a detailed description of the fine geometry of…

We begin by revisiting a paper of Erd\H{o}s and Fishburn, which posed the following question: given $k\in \mathbb{N}$, what is the maximum number of points in a plane that determine at most $k$ distinct distances, and can such optimal…

We prove effective joint equidistribution of several natural parameters associated to primitive vectors in $\mathbb{Z}^{n}$, as the norm of these vectors tends to infinity. These parameters include the direction, the orthogonal lattice, and…

For $g < n$, let $b\_1,...,b\_{n-g}$ be $n - g$ independent vectors in $\mathbb{R}^n$ with a common distribution invariant by rotation. Considering these vectors as a basis for the Euclidean lattice they generate, the aim of this paper is…

Let $L\subset \mathbb{Z}^n$ be a lattice and $I_L=\langle x^{\bf u}-x^{\bf v}:\ {\bf u}-{\bf v}\in L\rangle$ be the corresponding lattice ideal in $\Bbbk[x_1,\ldots, x_n]$, where $\Bbbk$ is a field. In this paper we describe minimal…

We study bond percolation on several four-dimensional (4D) lattices, including the simple (hyper) cubic (SC), the SC with combinations of nearest neighbors and second nearest neighbors (SC-NN+2NN), the body-centered cubic (BCC), and the…

We construct families of symmetric, antisymmetric, and asymmetric solitary modes in one-dimensional bichromatic lattices with the second-harmonic-generating ($\chi ^{(2)}$) nonlinearity concentrated at a pair of sites placed at distance…

We prove a highly uniform stability or "almost-near" theorem for dual lattices of lattices $L \subseteq \Bbb R^n$. More precisely, we show that, for a vector $x$ from the linear span of a lattice $L \subseteq \Bbb R^n$, subject to…