Related papers: On the growth of multi-recurrences

In this paper we introduce a unified approach to deal with incidence problems between points and varieties over finite fields. More precisely, we prove that the number of incidences $I(\mathcal{P}, \mathcal{V})$ between a set $\mathcal{P}$…

We give an exponential improvement to the diagonal van der Waerden numbers for $r\ge 5$ colors.

We gave an alternative short proof on the finite generation of holomorphic functions with polynomial growth on Riemann surfaces with nonnegative curvature. The first proof was due to Li and Tam.

We give a formula and an estimation for the number of irreducible polynomials in two (or more) variables over a finite field.

Suppose that $G_j$ is a sequence of finite connected planar graphs, and in each $G_j$ a special vertex, called the root, is chosen randomly-uniformly. We introduce the notion of a distributional limit $G$ of such graphs. Assume that the…

We examine the polynomial analogues of McMullen's and Zaremba's conjectures on continued fractions with bounded partial quotients. It has already been proved by Blackburn that if the base field is infinite, then the polynomial analogue of…

In this article, we provide upper and lower bounds for the growth rate of irreducible meanders. The obtained upper bound implies that the proportion of irreducible meanders among all of the prime meanders of order $n$ approaches $0$ as $n$…

We show how a recent result by Aka and Shapira on the evolution of continued fractions in a fixed quadratic field implies the classic result of de Mathan and Teulli\'e on the Mixed Littlewood

Given a finite grid in $\mathbb{R}^2$, how many lines are needed to cover all but one point at least $k$ times? Problems of this nature have been studied for decades, with a general lower bound having been established by Ball and Serra. We…

We improve upon the upper bounds for the cardinality of the value set of a multivariable polynomial map over a finite field using the polytope of the polynomial. This generalizes earlier bounds only dependent on the degree of a polynomial.

This letter is a comment on an article by T.C. Halsey and M.H. Jensen in Nature about using recurrence times as a reliable tool to estimate multifractal dimensions of strange attractors. Our aim is to emphasize that in the recent…

We show that given any tiling of Euclidean space, any geometric patterns of points, we can find a patch of tiles (of arbitrarily large size) so that copies of this patch appear in the tiling nearly centered on a scaled and translated…

We establish results with an arithmetic flavor that generalize the polynomial multidimensional Szemeredi theorem and related multiple recurrence and convergence results in ergodic theory. For instance, we show that in all these statements…

We prove an improvement on Schmidt's upper bound on the number of number fields of degree $n$ and absolute discriminant less than X for $6 \leq n \leq 94$. We carry this out by improving and applying a uniform bound on the number of monic…

We prove an upper bound for the length of an arithmetic progression represented by an irreducible integral binary quadratic form or a norm form, which depends only on the form and the progression's common difference. For quadratic forms,…

We introduce a new natural notion of convergence for permutations at any specified scale, in terms of the density of patterns of restricted width. In this setting we prove that limits may be chosen independently at a countably infinite…

We study proportions of consecutive occurrences of permutations of a given size. Specifically, the feasible limits of such proportions on large permutations form a region, called feasible region. We show that this feasible region is a…

We prove that any finitely generated one ended group has linear end depth. Moreover, we give alternative proofs to theorems relating the growth of a finitely generated group to the number of its ends.

The notion of the capacity of a polynomial was introduced by Gurvits around 2005, originally to give drastically simplified proofs of the Van der Waerden lower bound for permanents of doubly stochastic matrices and Schrijver's inequality…

We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of J. von zur Gathen and I. E. Shparlinski.