arXiv subjects
Publications and source records attributed to Alan Haynes.
This paper addresses a problem recently raised by Laurent and Nogueira about inhomogeneous Diophantine approximation with coprime integers. As a corollary of our main theorem we obtain an improvement of the best known exponent of approximation in this problem, from 1/2 to 1-epsilon, for any epsilon>0.
This paper takes a new step in the direction of proving the Duffin-Schaeffer Conjecture for measures arbitrarily close to Lebesgue. The main result is that under a mild `extra divergence' hypothesis, the conjecture is true.
Under suitable hypotheses we establish a quantitative pointwise ergodic theorem which applies to trimmed Birkhoff sums of weakly integrable functions.
In this paper we consider the probabilistic theory of Diophantine approximation in projective space over a completion of Q. Using the projective metric studied by Bombieri, van der Poorten, and Vaaler we prove the analogue of Khintchine's Theorem in projective space. For finite places and in higher dimension, we are able to completely remove the condition of monotonicity and establish the analogue of the Duffin-Schaeffer conjecture.
In this paper we study the Mixed Littlewood Conjecture with pseudo-absolute values. We show that if p is a prime and D is a pseudo-absolute value sequence satisfying mild conditions then then the infimum over natural numbers n of the quantity n.|n|_p.|n|_D.||nx|| equals 0 for all real x. Our proof relies on a measure rigidity theorem due to Lindenstrauss and lower bounds for linear forms in logarithms due to Baker and Wustholz. We also deduce the answer to the related metric question of how fast the infimum above tends to zero, for almost every x.
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 motivated by the problem of constructing strict deformation quantizations of symplectic manifolds. We show that this type of approximation exists for any real number and also investigate what happens if the number is rational or a quadratic irrational.
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 result is that, for almost every real number α, \liminf_{n\rar\infty}n(\log n)^2|n|_p\|nα\|=0.
Given a nonnegative function $ψ: \N \to \R $, let $W(ψ)$ denote the set of real numbers $x$ such that $|nx -a| < ψ(n) $ for infinitely many reduced rationals $a/n (n>0) $. A consequence of our main result is that $W(ψ)$ is of full Lebesgue measure if there exists an $ε> 0 $ such that $$ \textstyle \sum_{n\in\N}(\frac{ψ(n)}{n})^{1+ε}φ(n)=\infty . $$ The Duffin-Schaeffer Conjecture is the corresponding statement with $ε= 0$ and represents a fundamental unsolved problem in metric number theory. Another consequence is that $W(ψ)$ is of full Hausdorff dimension if the above sum with $ε= 0$ diverges; i.e. the dimension analogue of the Duffin-Schaeffer Conjecture is true.