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12 papers · 1 filter
A note on Farey fractions with denominators in arithmetic progressions
Dmitry A. Badziahin, Alan K. Haynes
As the conclusion of a line of investigation undertaken in two previous papers, we compute asymptotic frequencies for the values taken by numerators of differences of consecutive F…
Numerators of differences of nonconsecutive Farey fractions
Alan K. Haynes
An elementary but useful fact is that the numerator of the difference of two consecutive Farey fractions is equal to one. For triples of consecutive fractions the numerators of the…
Intermediate convergents and a metric theorem of Khinchin
Alan K. Haynes
A landmark theorem in the metric theory of continued fractions begins this way: Select a non-negative real function defined on the positive integers and a real number , and…
The metric theory of p-adic approximation
Alan K. Haynes
Metric Diophantine approximation in its classical form is the study of how well almost all real numbers can be approximated by rationals. There is a long history of results which g…
The Duffin-Schaeffer Conjecture with extra divergence
Alan Haynes, Andrew Pollington, Sanju Velani
Given a nonnegative function , let denote the set of real numbers such that for infinitely many reduced rationals . A conseq…
Inhomogeneous Diophantine approximation on curves and Hausdorff dimension
Dzmitry Badziahin
The goal of this paper is to develop a coherent theory for inhomogeneous Diophantine approximation on curves in akin to the well established homogeneous theory. More specific…