Lattice convex chains in the plane

Let $G$ be an infinite, vertex-transitive lattice with degree $\lambda$ and fix a vertex on it. Consider all cycles of length exactly $l$ from this vertex to itself on $G$. Erasing loops chronologically from these cycles, what is the fraction $F_p/\lambda^{\ell(p)}$ of cycles of length $l$ whose last erased loop is some chosen self-avoiding polygon $p$ of length $\ell(p)$, when $l\to\infty$ ? We use combinatorial sieves to prove an exact formula for $F_p/\lambda^{\ell(p)}$ th...

We prove area bounds for planar convex bodies in terms of their number of interior integral points and their lattice width data. As an application, we obtain sharp area bounds for rational polygons with a fixed number of interior integral points depending on their denominator. For lattice polygons, we also present an equation for the area based on Noether's formula.

We provide a new strategy to compute the exponential growth constant of enumeration sequences counting walks in lattice path models restricted to the quarter plane. The bounds arise by comparison with half-planes models. In many cases the bounds are provably tight, and provide a combinatorial interpretation of recent formulas of Fayolle and Raschel (2012) and Bostan, Raschel and Salvy (2013). We discuss how to generalize to higher dimensions.

Assume $K$ is a convex body in $R^d$, and $X$ is a (large) finite subset of $K$. How many convex polytopes are there whose vertices come from $X$? What is the typical shape of such a polytope? How well the largest such polytope (which is actually $\conv X$) approximates $K$? We are interested in these questions mainly in two cases. The first is when $X$ is a random sample of $n$ uniform, independent points from $K$ and is motivated by Sylvester's four-point problem, and by th...

Under the Riemann Hypothesis, we improve the error term in the asymptotic formula related to the counting lattice problem studied in a first part of this work. The improvement comes from the use of Weyl's bound for exponential sums of polynomials and a device due to Popov allowing us to get an improved main term in the sums of certain fractional parts of polynomials.

Let $P$ be a finite point set in $\mathbb{R}^2$ with the set of distance $n$-chains defined as $$ \Delta_n(P)=\{(|p_1-p_2|,|p_2-p_3|,\ldots,|p_n-p_{n+1}|):p_i \in P\}.$$ We show that for $2\leq n=O_{|P|}(1)$ we have $$|\Delta_n(P)|\gtrsim \frac{|P|^{n}}{\log^{\frac{13}{2}(n-1)}|P|}.$$ Our argument uses the energy construction of Elekes and a general version of Rudnev's rich-line bound implicit in Rudnev's recent hinge paper which allows one to iterate efficiently on highly in...

This paper is a continuation of an earlier one, and completes a classification of the configurations of points in a plane lattice that determine angles that are rational multiples of ${\pi}$. We give a complete and explicit description of lattices according to which of these configurations can be found among their points.

In this paper we study the classification problem of convex lattice ploytopes with respect to given volume or given cardinality.

In this paper, we enumerate Newton polygons asymptotically. The number of Newton polygons is computable by a simple recurrence equation, but unexpectedly the asymptotic formula of its logarithm contains growing oscillatory terms. As the terms come from non-trivial zeros of the Riemann zeta function, an estimation of the amplitude of the oscillating part is equivalent to the Riemann hypothesis.

Counting integer points in large convex bodies with smooth boundaries containing isolated flat points is oftentimes an intermediate case between balls (or convex bodies with smooth boundaries having everywhere positive curvature) and cubes (or convex polytopes). In this paper we provide a detailed description of several discrepancy problems in the particular planar case where the boundary coincides locally with the graph of the function $\mathbb{R\ni}t\mapsto\left\vert t\righ...