Related papers: On some open problems in Diophantine approximation
We prove new results, related to the Littlewood and Mixed Littlewood conjectures in Diophantine approximation.
We collect a number of open questions concerning Diophantine equations, Diophantine Approximation and transcendental numbers. Revised version: corrected typos and added references.
We survey classical and recent results on exponents of Diophantine approximation. We give only a few proofs and highlight several open problems.
We first propose two conjectural estimates on Diophantine approximation of logarithms of algebraic numbers. Next we discuss the state of the art and we give further partial results on this topic.
A basic question of Diophantine approximation, which is the first issue we discuss, is to investigate the rational approximations to a single real number. Next, we consider the algebraic or polynomial approximations to a single complex…
We discuss some easy statements dealing with linear inhomogeneous Diophantine approximation. Surprisingly, we did not find some of them in the literature.
We introduce an inhomogeneous variant of Kaufman's measure, with applications to diophantine approximation. In particular, we make progress towards a problem related to Littlewood's conjecture.
This paper collects polynomial Diophantine equations that are simple to state but apparently difficult to solve.
The Littlewood Conjecture in Diophantine approximation can be thought of as a problem about covering the plane by a union of hyperbolas centered at rational points. In this paper we consider the problem of translating the center of each…
We discuss the problem of finding optimal exponents in Diophantine estimates involving one real number and, in some cases where such an exponent is known, present some properties of the corresponding extremal numbers.
We consider some asymptotic analysis for series related to the work of Hardy and Littlewood on Diophantine approximation, as well as Davenport. In particular, we expand on ideas from some previous work on arithmetic series and the RH.
The main goal of this note is to develop a metrical theory of Diophantine approximation within the framework of the de Mathan-Teulie Conjecture, also known as the `Mixed Littlewood Conjecture'. Let p be a prime. A consequence of our main…
This paper reports on the current status of the project in which we order all polynomial Diophantine equations by an appropriate version of "size", and then solve the equations in that order. We list the "smallest" equations that are…
We study the problem of Diophantine approximation on lines in $\mathbb{R}^d$ under certain primality restrictions.
In this paper we adopt a geometric point of view regarding a famous conjecture due to Littlewood in diophantine approximation of real numbers. Following the spirit of the geometric theory of continued fractions, we give a sufficient…
We consider the problem of Diophantine approximation on semisimple algebraic groups by rational points with restricted numerators and denominators and establish a quantitative approximation result for all real points in the group by…
We establish a strong form of Littlewood's conjecture with inhomogeneous shifts, for a full-dimensional set of pairs of badly approximable numbers on a vertical line. We also prove a uniform assertion of this nature, generalising a strong…
This paper is motivated by two problems in the theory of Diophantine approximation, namely, Davenport's problem regarding badly approximable points on submanifolds of a Euclidean space and Schmidt's problem regarding the intersections of…
In a previous paper, we studied certain sequences of simultaneous rational approximations in ${\bf R}^2$ which present some analogy with the continued fractions. We got results around the Littlewood conjecture by using such approximations.…
We study some problems in metric Diophantine approximation over local fields of positive characteristic.