Related papers: Explicit bounds for rational points near planar cu…
A point (x1, x2) with coordinates in a subfield of R of transcendence degree one over Q, with 1, x1, x2 linearly independent over Q, may have a uniform exponent of approximation by elements of Q^2 that is strictly larger than the lower…
Markov's theorem classifies the worst irrational numbers with respect to rational approximation and the indefinite binary quadratic forms whose values for integer arguments stay farthest away from zero. The main purpose of this paper is to…
We refine upper bounds for the classical exponents of uniform approximation for a linear form on the Veronese curve in dimension from $3$ to $9$. For dimension three, this in particular shows that a bound previously obtained by two…
In this article, we study an analytic curve $\varphi: I=[a,b]\rightarrow \mathrm{M}(n\times n, \mathbb{R})$ in the space of $n$ by $n$ real matrices, and show that if $\varphi$ satisfies certain geometric conditions, then for almost every…
We prove a refined version of Markov's theorem in Diophantine approximation. More precisely, we characterize completely the set of irrationals $x$ such that $\left|x-\frac{p}{q}\right|<\frac{1}{3q^2}$ has only finitely many rational…
We obtain optimal boundary and global regularity estimates for viscosity solutions of fully nonlinear elliptic equations whose ellipticity degenerates at the critical points of a given solution. We show that any solution is $C^{1,\alpha}$…
We develop a new framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when working with real-world…
A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a…
We establish arithmetical properties and provide essential bounds for bi-sequences of approximation coefficients associated with the natural extension of maps, leading to continued fraction-like expansions. These maps are realized as the…
The notions of bounded expansion and nowhere denseness have been applied very successfully in algorithmic graph theory. We study the corresponding notions of directed bounded expansion and nowhere crownfulness on directed graphs. We show…
In this article we prove the explicit Mordell Conjecture for large families of curves. In addition, we introduce a method, of easy application, to compute all rational points on curves of quite general shape and increasing genus. The method…
We establish a new upper bound for the number of rationals up to a given height in a missing-digit set, making progress towards a conjecture of Broderick, Fishman, and Reich. This enables us to make novel progress towards another conjecture…
We continue the development, by reduction to a first order system for the conormal gradient, of $L^2$ \textit{a priori} estimates and solvability for boundary value problems of Dirichlet, regularity, Neumann type for divergence form second…
We give elementary proof of stronger versions of several recent results on intrinsic Diophantine approximation on rational quadric hypersurfaces $X\subset \mathbb{P}^n(\mathbb{R})$. The main tool is a refinement of the simplex lemma, which…
In 1922, Mordell conjectured that the set of rational points on a smooth curve $C$ over $\mathbb{Q}$ with genus $g \ge 2$ is finite. This has been proved by Faltings in 1983. However, Coleman determined in 1985 an upper bound of…
A general form of the Borel-Cantelli Lemma and its connection with the proof of Khintchine's Theorem on Diophantine approximation and the more general Khintchine-Groshev theorem are discussed. The torus geometry in the planar case allows a…
It is well known since Noether that the gonality of a smooth plane curve of degree d>3 is d-1. Given a k-dimensional complex projective variety X, the most natural extension of gonality is probably the degree of irrationality, that is the…
We study the diophantine exponent of analytic submanifolds of the space of m by n real matrices, answering questions of Beresnevich, Kleinbock and Margulis. We identify a family of algebraic obstructions to the extremality of such a…
We consider rational points on the sphere and investigate their equidistribution in shrinking spherical caps. For the two-dimensional sphere, we leverage Hecke operators to obtain a significantly improved small-scale equidistribution bound,…
The theory of classical types of curves in normed planes is not strongly developed. In particular, the knowledge on existing concepts of curvatures of planar curves is widespread and not systematized in the literature. Giving a…