Related papers: On some open problems in Diophantine approximation
We give several results related to inhomogeneous approximations to two real numbers and badly approximable numbers. Our results are related to classical theorems by A. Khintchine (1926) and to an original method invented by Y. Peres and W.…
We estimate the lattice sums arising in the context of the integer point counting in polyhedra.
In this paper we give a survey of what is currently known about Diophantine exponents of lattices and propose several problems.
We study the problem of Diophantine approximation on lines in R^2 with prime numerator and denominator.
Diophantine approximation is the problem of approximating a real number by rational numbers. We propose a version of this in which the numerators are approximately related to the denominators by a Laurent polynomial. Our definition is…
A long-standing conjecture of Littlewood about simultaneous Diophantine approximation has an analogous problem for a field of formal Laurent series $\mathbb{F}(\!(t^{-1})\!)$. That is, we can ask whether for any series $\Theta$, $\Phi$ and…
A detailed survey of the Lonely Runner Conjecture and its connection between Diophantine approximation and View-obstruction problems.
This note pushes further the discussion about relations between Dirichlet improvable, badly approximable and singular points held in recent joint work with Beresnevich, Guan, Velani and Ramirez, by considering Diophantine sets extending the…
We refine a result of the last two Authors of [8] on a Diophantine approximation problem with two primes and a $k$-th power of a prime which was only proved to hold for $1<k<4/3$. We improve the $k$-range to $1<k\le 3$ by combining Harman's…
In these notes, we begin by recalling aspects of the classical theory of metric Diophantine approximation; such as theorems of Khintchine, Jarn\'{\i}k, Duffin-Schaeffer and Gallagher. We then describe recent strengthening of various…
We establish a `mixed' version of a fundamental theorem of Khintchine within the field of simultaneous Diophantine approximation. Via the notion of ubiquity we are able to make significant progress towards the completion of the metric…
We prove that the Littlewood conjecture is satisfied for a restricted class of pairs $(\alpha,\beta)$ of badly approximable numbers. We use the localization of the roots of a cubic equation with coefficients depending on the diophantine…
We give some comments on W.M. Schmidt's theorem on Diophantine approximations with positive integers and our recent results on the topic.
This brief survey deals with multi-dimensional Diophantine approximations in sense of linear form and with simultaneous Diophantine approximations. We discuss the phenomenon of degenerate dimension of linear subspaces generated by the best…
In this paper we give solutions of certain diophantine equations related to triangular and tetrahedral numbers and propose several problems connected with these numbers. The material of this paper was presented in part at the 11th…
In this note we briefly survey and propose some open problems related to isoparametric theory.
Recent developments in the theory and application of the Hardy-Littlewood method are discussed, concentrating on aspects associated with diagonal diophantine problems. Recent efficient differencing methods for estimating mean values of…
We present in this article a general approach (in the form of recommendations and guidelines) for tackling Diophantine equation problems (whether single equations or systems of simultaneous equations). The article should be useful in…
Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are…
We investigate approximation to a given real number by algebraic numbers and algebraic integers of prescribed degree. We deal with both best and uniform approximation, and highlight the similarities and differences compared with the…