Related papers: Primitive point packing

We prove a sharp bound for the remainder term of the number of lattice points inside a ball, when averaging over a compact set of (not necessarily unimodular) lattices, in dimensions two and three. We also prove that such a bound cannot…

We propose a unified framework that synthesizes advances in high-dimensional lattice theory with novel computational algorithms for the shortest vector problem (SVP) to model pure root lattices and compute sphere packing densities. Building…

A set $\mathcal{A}\subset \mathbb{N}$ is called additively decomposable (resp. asymptotically additively decomposable) if there exist sets $\mathcal{B},\mathcal{C}\subset \mathbb{N}$ of cardinality at least two each such that…

For a finite set of integers such that the first few gaps between its consecutive elements equal $a$, while the remaining gaps equal $b$, we study dense packings of its translates on the line. We obtain an explicit lower bound on the…

Global in time well-posedness of $H^2$ solutions and $z$-weak solutions of the 3D Primitive equations in a bounded cylindrical domain is proved. More specifically, uniform in time boundedness and bounded absorbing sets are obtained for both…

We set out an elementary approach to derive Visible Point Identities summed on lattice points of inverted triangle (2D), pyramid (3D), hyperpyramid (4D, 5D and so on) utilizing the greatest common divisor for the nD Visible Point Vectors.…

We develop novel techniques using abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinite-volume hyperbolic manifolds, with error terms which are uniform as the lattice moves through "congruence"…

We ask whether a stationary lattice in dimension $d$ whose points are shifted by identically distributed but possibly dependent perturbations remains hyperuniform. When $d = 1$ or $2$, we show that it is the case when the perturbations have…

Given a polynomial f(z) = z^d + c over a global field K and a_0 in K, we study the density of prime ideals of K dividing at least one element of the orbit of a_0 under f. The density of such sets for linear polynomials has attracted much…

We show that if a big set of integer points in [0,N]^d, d>1, occupies few residue classes mod p for many primes p, then it must essentially lie in the solution set of some polynomial equation of low degree. This answers a question of…

A set is primitive if no element of the set divides another. We consider primitive sets of monic polynomials over a finite field and find natural generalizations of many of the results known for primitive sets of integers. In particular we…

Let $d$ and $k$ be integers with $1 \leq k \leq d-1$. Let $\Lambda$ be a $d$-dimensional lattice and let $K$ be a $d$-dimensional compact convex body symmetric about the origin. We provide estimates for the minimum number of $k$-dimensional…

Among integral polytopes (vertices with integral coordinates), lattice-free polytopes - intersecting the lattice ONLY at their vertices- are of particular interestin combinatorics and geometry of numbers. A natural question is to measure…

For natural numbers $n$ and $l > d \geq 2$, let $ES_d(l,n)$ be the minimum $N$ such that any set of at least $N$ points in $\mathbb{R}^d$ contains either $l$ points contained in a common $(d-1)$-dimensional hyperplane or $n$ points in…

Let $\mathcal{P} \subset \mathbb{R}^d$ be a lattice polytope of dimension $d$. Let $b(\mathcal{P})$ denote the number of lattice points belonging to the boundary of $\mathcal{P}$ and $c(\mathcal{P})$ that to the interior of $\mathcal{P}$.…

Consider the integer best approximations of a linear form in $n\ge 2$ real variables. While it is well-known that any tail of this sequence always spans a lattice is sharp for any $n\ge 2$. In this paper, we determine the exact Hausdorff…

We prove the existence of primitive sets (sets of integers in which no element divides another) in which the gap between any two consecutive terms is substantially smaller than the best known upper bound for the gaps in the sequence of…

A set of natural numbers $A$ is called primitive if no element of $A$ divides any other. Let $\Omega(n)$ be the number of prime divisors of $n$ counted with multiplicity. Let $f_z(A) = \sum_{a \in A}\frac{z^{\Omega(a)}}{a (\log a)^z}$,…

Given $d\geq 2$, we show that the number of approximates $\frac{1}{q}\mathbf{p}\in \mathbb{Q}^d$ of $\mathbf{x}\in\mathbb{R}^d$ satisfying $|q\mathbf{x}-\mathbf{p}|\leq cq^{-\frac{1}{d}}$ with denominator $1\leq q < T$ decays to the…

We associate to lattice points a_0,a_1,...,a_N in Z^n an A-hypergeometric series \Phi(\lambda) with integer coefficients. If a_0 is the unique interior lattice point of the convex hull of a_1,...,a_N, then for every prime p\neq 2 the ratio…