Related papers: The average number of integral points in orbits

The mapping class group of a surface $\S$ acts on the set of closed geodesics on $\S$. This action preserves self-intersection number. In this paper, we count the orbits of curves with at most $K$ self-intersections, for each $K \geq 1$.…

In this paper we study the geometry of metric spheres in the curve complex of a surface, with the goal of determining the "average" distance between points on a given sphere. Averaging is not technically possible because metric spheres in…

For each $n=1,2,\dots$, let $\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$ be the affine group over the integers. For every point $x=(x_1,\dots,x_n) \in \mathbb{R}^n$ let $\mathrm{orb}(x)=\{\gamma(x)\in \mathbb{R}^n\mid\gamma\in…

We prove the first inverse theorem for point--sphere incidence bounds over finite fields in dimensions $d \ge 3$, showing that near-extremality forces algebraic rigidity. While sharp upper bounds have been known for over a decade, the…

Let $\mathbb{K}$ be a field of characteristic zero and $\mathbb{K}[x_1, \dots, x_n]$ the corresponding multivariate polynomial ring. Given a sequence of $s$ polynomials $\mathbf{f} = (f_1, \dots, f_s)$ and a polynomial $\phi$, all in…

Given a polynomial ring $P$ over a field $K$, an element $g \in P$, and a $K$-subalgebra $S$ of $P$, we deal with the problem of saturating $S$ with respect to $g$, i.e. computing $Sat_g(S) = S[g, g^{-1}]\cap P$. In the general case we…

There are well-known examples of dynamical systems for which the Birkhoff averages with respect to a given observable along some or all of the orbits do not converge. It has been suggested that such orbits could be classified using higher…

The infinitesimal space of a quasiregular mapping was introduced by Gutlyanskii et al and generalized the idea of a derivative for this class of mappings which is only differentiable almost everywhere. In this paper, we show that the…

We prove that zeros and critical points of a random polynomial $p_N$ of degree $N$ in one complex variable appear in pairs. More precisely, if $p_N$ is conditioned to have $p_N(\xi)=0$ for a fixed $\xi \in \C\backslash\set{0},$ we prove…

In this article, we extend the Bufetov pointwise ergodic theorem for spherical averages of even radius for free group actions on noncommutative $L\log L$-space. Indeed, we extend it to more general Orlicz space $L^\Phi(M, \tau)$…

In this paper we study the action of the fundamental group of a finite metric graph on its universal covering tree. We assume the graph is finite, connected and the degree of each vertex is at least three. Further, we assume an…

A cap of spherical radius $\alpha$ on a unit $d$-sphere $S$ is the set of points within spherical distance $\alpha$ from a given point on the sphere. Let $\mathcal F$ be a finite set of caps lying on $S$. We prove that if no hyperplane…

Consider the integer points lying on the sphere of fixed radius projected onto the unit sphere. Duke showed that, on congruence conditions for the radius squared, these points equidistribute. To further this study of equidistribution, we…

We study the combinatorial types of periodic orbits of the standard covering endomorphisms ${\mathbf m}_k(x)=k x \ (\text{mod} \ {\mathbb Z})$ of the circle for integers $k \geq 2$ and the frequency with which they occur. For any $q$-cycle…

We obtain a general lower bound for the number of fixed points of a circle action on a compact almost complex manifold $M$ of dimension $2n$ with nonempty fixed point set, provided the Chern number $c_1c_{n-1}[M]$ vanishes. The proof…

For a linear group $G$ acting on an absolutely irreducible variety $X$ over the rationals $\QQ$, we describe the orbits of $X(\QQ_p)$ under $G(\QQ_p)$ and of $X(\FF_p((t)))$ under $G(\FF_p((t)))$ for $p$ big enough. This allows us to show…

This paper is about the existence of periodic orbits near an equilibrium point of a two-degree-of-freedom Hamiltonian system. The equilibrium is supposed to be a nondegenerate minimum of the Hamiltonian. Every sphere-like component of the…

The present paper analyzes the discrepancy of distribution of rational points on general semisimple algebraic group varieties. The results include mean-square, almost sure, and uniform discrepancy estimates with explicit error bounds, which…

We show that if $f : \mathbb{A}_{\bar{\mathbb{Q}}}^r \to \mathbb{A}_{\bar{\mathbb{Q}}}^r$ is a regular self-map and $P \in \mathbb{A}^r(\bar{\mathbb{Q}})$ has $\limsup_{n \in \mathbb{N}} \frac{\log{h_{\mathrm{aff}}(f^nP)}}{\log{n}} < 1/r$,…

We study $\kappa$-maximal cofinitary groups for $\kappa$ regular uncountable, $\kappa = \kappa^{<\kappa}$. Revisiting earlier work of Kastermans and building upon a recently obtained higher analogue of Bell's theorem, we show that: 1. Any…