Lattice convex chains in the plane

The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states that for any sequence $u_1,\dots, u_n$ of norm 1 vectors in a real Hilbert space $\mathscr H$, there exists a unit vector $v \in \mathscr H$, such that $$ \sum \frac{1}{\langle u_i, v \rangle^2} \leq n^2. $$ The 2-dimensional case is proved by complex analytic m...

Let $\gamma$ be a bounded convex curve on a plane. Then $\sharp (\gamma\cap (\Z/n)^2)=o(n^{2/3})$. It streghtens the classical result of Jarn\'\i k (an upper estimate $O(n^{2/3})$) and disproves a conjecture of Vershik on existence of the so-called {\it universal Jarn\'\i k curve}.

Let $\Delta \subset \R^n$ be an $n$-dimensional lattice polytope. It is well-known that $h_{\Delta}^*(t) := (1-t)^{n+1} \sum_{k \geq 0} |k\Delta \cap \Z^n| t^k $ is a polynomial of degree $d \leq n$ with nonnegative integral coefficients. Let $AGL(n, \Z)$ be the group of invertible affine integral transformations which naturally acts on $\R^n$. For a given polynomial $h^* \in \Z[t]$, we denote by $C_{h^*}(n)$ the number $AGL(n, \Z)$-equivalence classes of $n$-dimensional latt...

We solve a randomized version of the following open question: is there a strictly convex, bounded curve \gamma in the plane such that the number of rational points on \gamma, with denominator $n$, approaches infinity with $n$? Although this natural problem appears to be out of reach using current methods, we consider a probabilistic analogue using a spatial Poisson-process that simulates the refined rational lattice $\frac{1}{d} Z^2$, which we call $M_d$, for each natural num...

Let $\mathbb{P}_{\kappa}(n)$ be the probability that $n$ points $z_1,\ldots,z_n$ picked uniformly and independently in $\mathfrak{C}_\kappa$, a regular $\kappa$-gon with area $1$, are in convex position, that is, form the vertex set of a convex polygon. In this paper, we give an equivalent of $\mathbb{P}_{\kappa}(n)$ for all $\kappa\geq 3$, which improves on a famous result of B\'ar\'any. A second aim of the paper is to establish a limit theorem which describes the fluctuatio...

The Kirchhoff index is defined as the sum of resistance distances between all pairs of vertices in a graph. This index is a critical parameter for measuring graph structures. In this paper, we characterize polygonal chains with the minimum Kirchhoff index, and characterize even (odd) polygonal chains with the maximum Kirchhoff index, which extends the results of \cite{45}, \cite{14} and \cite{2,13,44} to a more general case.

In this paper we investigate the number of integer points lying in dilations of lattice path matroid polytopes. We give a characterization of such points as polygonal paths in the diagram of the lattice path matroid. Furthermore, we prove that lattice path matroid polytopes are affinely equivalent to a family of distributive polytopes. As applications we obtain two new infinite families of matroids verifying a conjecture of De Loera et.~al. and present an explicit formula of ...

In this paper for any dimension n we give a complete list of lattice convex polytopes in R^n that are regular with respect to the group of affine transformations preserving the lattice.

We prove explicit bounds on the number of lattice points on or near a convex curve in terms of geometric invariants such as length, curvature, and affine arclength. In several of our results we obtain the best possible constants. Our estimates hold for lattices more general than the usual lattice of integral points in the plane.

This paper is concerned with configurations of points in a plane lattice which determine angles that are rational multiples of $\pi$. We shall study how many such angles may appear in a given lattice and in which positions, allowing the lattice to vary arbitrarily. This classification turns out to be much less simple than could be expected, leading even to parametrizations involving rational points on certain algebraic curves of positive genus.Bulletin of the American Mathema...