Related papers: Simultaneous visibility in the algebraic lattice

An integral quadratic lattice is called indefinite $k$-universal if it represents all integral quadratic lattices of rank $k$ for a given positive integer $k$. For $k\geq 3$, we prove that the indefinite $k$-universal property satisfies the…

In this article we prove that the co-compactness of the arithmetic lattices in a connected semisimple real Lie group is preserved if the lattices under consideration are representation equivalent. This is in the spirit of the question posed…

We discuss principality of prime ideals of finite algebraic number fields $L=K(\theta)$ over an algebraic number field $K ([K:\mathbb{Q}]<\infty)$ defined by irreducible polynomials $f(x)\in \mathfrak{O}_{K}[x]$ and $f(\theta)=0$. Our main…

We determine, asymptotically, the number of quintic fields having bounded discriminant. Specifically, we prove that the asymptotic number of quintic fields having absolute discriminant at most X is a constant times X. In contrast with the…

Based on a fairly precise approximation to the lattice discrepancy of a Lame disc, an asymptotic formula is established for the number of lattice points in a related three-dimensional body, linearly dilated by a large real parameter x.…

Let $X$ be an Enriques surface defined over a number field $K$. Then there exists a finite extension $K'/K$ such that the set of $K'$-rational points of $X$ is Zariski dense.

A vertical 2-sum of a two-coatom lattice $L$ and a two-atom lattice $U$ is obtained by removing the top of $L$ and the bottom of $U$, and identifying the coatoms of $L$ with the atoms of $U$. This operation creates one or two nonisomorphic…

We study the regularity and the algebraic properties of certain lattice ideals. We establish a map I --> I\~ between the family of graded lattice ideals in an N-graded polynomial ring over a field K and the family of graded lattice ideals…

For $n=3$, 4, and 5, we prove that, when $S_n$-number fields of degree $n$ are ordered by their absolute discriminants, the lattice shapes of the rings of integers in these fields become equidistributed in the space of lattices.

Let $K\subseteq \mathbb{R}$ be a number field. Using techniques of discrete analysis, we prove that for definable sets $X$ in $\mathbb{R}_{\exp}$ of dimension at most $2$ a conjecture of Wilkie about the density of rational points is…

Suppose that $L, M$ are two full-rank lattices in Euclidean space with $\text{vol}(L)=\text{vol}(M)$. We give a new proof on the existence of a bounded and Lebesgue measurable set that tiles $\mathbb{R}^d$ with both $L,M$ using the…

In this paper we shall consider the Lie algebra of column-finite infinite matrices indexed by positive integers $\mathbb{N}$, describe the lattice of its ideals for arbitrary field $K$ and study its derivations over any commutative, unital…

The "visibility" of a planar set $S$ from a point $a$ is defined as the normalized size of the radial projection of $S$ from $a$ to the unit circle centered at $a$. Simon and Solomyak (Real Anal. Exchange 2006/07) proved that unrectifiable…

Classification of differential-difference equation of the form $\ddot{u}_{nm}=F_{nm}\big(t, \{u_{pq}\}|_{(p,q)\in \Gamma}\big)$ are considered according to their Lie point symmetry groups. The set $\Gamma$ represents the point $(n,m)$ and…

We show that the process of spontaneous symmetry breaking can trap a field theoretic system in a highly non-trivial state containing a lattice of domain walls. In one large compact space dimension, a lattice is inevitably formed. In two…

We prove the density hypothesis for congruence subgroups of an irreducible uniform lattice in $\mathrm{PSL}_2(\mathbb{R})^d$, extending previous results on the spherical density hypothesis to bound multiplicities of non-tempered…

Let $A$ be the product of an abelian variety and a torus over a number field $K$, and let $m$ be a positive integer. If $\alpha \in A(K)$ is a point of infinite order, we consider the set of primes $\mathfrak p$ of $K$ such that the…

This article introduces the notion of good labellings for asymptotic lattices in order to study joint spectra of quantum integrable systems from the point of view of inverse spectral theory. As an application, we consider a new spectral…

Point sets of number-theoretic origin, such as the visible lattice points or the $k$-th power free integers, have interesting geometric and spectral properties and give rise to topological dynamical systems that belong to a large class of…

Let $\mathcal{O}$ be a Dedekind domain whose field of fractions $K$ is a global field. Let $A$ be a finite-dimensional separable $K$-algebra and let $\Lambda$ be an $\mathcal{O}$-order in $A$. Let $n$ be a positive integer and suppose that…