Related papers: Lattice Points in Large Borel Sets and Successive …

We consider random lattices taken from the general symplectic ensemble and count the number of lattice points of a typical lattice in nested families $B_t$ of certain Borel sets. Our main result is that for almost every general symplectic…

Let A be a subset of positive relative upper density of P^d, the d-tuples of primes. We prove that A contains an affine copy of any finite set of lattice points E, as long as E is in general position in the sense that it has at most one…

We investigate the asymptotic behavior of the distribution of primitive lattice points in a symmetric Borel set $S_d\subset\mathbb R^d$ as $d$ goes to infinity, under certain volume conditions on $S_d$. Our main technique involves exploring…

We show that any finite $S \subset \mathbb{R}^d$ in general position has arbitrarily large supersets $T \supseteq S$ in general position with the property that $T$ contains no empty convex polygon, or hole, with $C_d$ points, where $C_d$ is…

Given two nonempty subsets $A, B$ of a group $G$, they are said to form a co-minimal pair if $A \cdot B = G$, and $A' \cdot B \subsetneq G$ for any $\emptyset \neq A' \subsetneq A$ and $A\cdot B' \subsetneq G$ for any $\emptyset \neq B'…

We show that, if the interior of a lattice d-polytope P contains at least one lattice point, then it contains a lattice point whose coefficient of asymmetry with respect to P is at most b for some number b depending on d only. As an…

Minkowski's second theorem on successive minima asserts that the volume of a 0-symmetric convex body K over the covolume of a lattice \Lambda can be bounded above by a quantity involving all the successive minima of K with respect to…

Let $n \in \mathbb{Z}_{\geq 3}.$ Given any Borel subset $A$ of $\mathbb{R}^n$ with finite and nonzero measure, we prove that the probability that the set of primitive points of a random full-rank unimodular lattice in $\mathbb{R}^n$ does…

We show that whenever a separable subset $S$ of a complete metric space $X$ admits a $d$-dimensional weak tangent field, the set $S$ is close to being $d$-dimensional in the following sense. Whenever $\mu$ is a Borel finite measure on $X$…

We show that the number $p\_d$ of non-similar perfect $d$-dimensional lattices satisfies eventually the inequalities$e^{d^{1-\epsilon}}<p\_d<e^{d^{3+\epsilon}}$ for arbitrary smallstrictly positive $\epsilon$.

We adapt an argument of Tao and Vu to show that if $\lambda_1\le\cdots\le\lambda_d$ are the successive minima of an origin-symmetric convex body $K$ with respect to some lattice $\Lambda<\mathbb{R}^d$, and if we set…

In this article we will show that for every natural $d$ and $n>1$ there exists a natural number $t$ such that for every $d$-dimensional simplicial complex $\mathcal{T}$ with vertices in $\mathbb{Z}^d$, the number of lattice points in the…

Motivated by Stanley's results in \cite{St02}, we generalize the rank of a partition $\lambda$ to the rank of a shifted partition $S(\lambda)$. We show that the number of bars required in a minimal bar tableau of $S(\lambda)$ is max$(o, e +…

Let $\# K$ be a number of integer lattice points contained in a set $K$. In this paper we prove that for each $d\in {\mathbb N}$ there exists a constant $C(d)$ depending on $d$ only, such that for any origin-symmetric convex body $K \subset…

We classify all tuples of lattice polyhedra of relative mixed volume 1 and all minimal (by inclusion) tuples of polyhedra of relative mixed volume 2. We also prove a conjecture by A. Esterov, which states that all tuples with finite…

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 $\mathcal{P}$.…

Let $G$ be a real centre-free semisimple Lie group without compact factors. I prove that irreducible lattices in $G$ are rigid under two types of sublinear distortions. The first result is that the class of lattices in groups that do not…

We prove that for any two lattices $L, M \subseteq \mathbb{R}^d$ of the same volume there exists a measurable, bounded, common fundamental domain of them. In other words, there exists a bounded measurable set $E \subseteq \mathbb{R}^d$ such…

Let S1(Gamma,Gamma) be the statement: For each sequence of point-cofinite open covers, one can pick one element from each cover and obtain a point-cofinite cover. b is the minimal cardinality of a set of reals not satisfying…

In this paper we study the set of Li-Yorke $d$-tuples and its $d$-dimensional Lebesgue measure for interval maps $T\colon [0,1] \to [0,1]$. If a topologically mixing $T$ preserves an absolutely continuous probability measure 9with respect…