arXiv subjects
Publications and source records attributed to Alan K. Haynes.
In this paper we examine the subset of Farey fractions of order Q consisting of those fractions whose denominators are odd. In particular, we consider the frequencies of values of numerators of differences of consecutive elements in this set. After proving an asymptotic result for these frequencies, we use estimates coming from incomplete Kloosterman sums to generalize our result to subintervals of [0,1].
In this paper we generalize some of our results from, `A note on Farey fractions with odd denominators' to subsets of Farey fractions consisting of fractions with denominators not divisible by a given prime. We also investigate the joint distribution of numerators of differences of h-tuples of consecutive fractions in these sets.
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 give partial answers to this problem, but there are still questions which remain unknown. The Duffin-Schaeffer Conjecture is an attempt to answer all of these questions in full, and it has withstood more than fifty years of mathematical investigation. In this paper we establish a strong connection between the Duffin-Schaeffer Conjecture and its p-adic analogue. Our main theorems are transfer principles which allow us to go back and forth between these two problems. We prove that if the variance method from probability theory can be used to solve the p-adic Duffin-Schaeffer Conjecture for even one prime p, then almost the entire classical Duffin-Schaeffer Conjecture would follow. Conversely if the variance method can be used to prove the classical conjecture then the p-adic conjecture is true for all primes. Furthermore we are able to unconditionally and completely establish the higher dimensional analogue of this conjecture in which we allow simultaneous approximation in any finite number and combination of real and p-adic fields, as long as the total number of fields involved is greater than one. Finally by using a mass transference principle for Hausdorff measures we are able to extend all of our results to their corresponding analogues with Haar measures replaced by the Hausdorff measures associated with arbitrary dimension functions.
First we prove some elementary but useful identities in the group ring of Q/Z. Our identities have potential applications to several unsolved problems which involve sums of Farey fractions. In this paper we use these identities, together with some analytic number theory and results about divisors in short intervals, to estimate the cardinality of a class of sets of fundamental interest.
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 differences are well understood and have applications to several interesting problems. In this paper we investigate numerators of differences of fractions which are farther apart. We establish algebraic identities between such differences which then allow us to calculate their average values by using properties of a measure preserving transformation of the Farey triangle.
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 Farey fractions with denominators restricted to lie in arithmetic progression.
A landmark theorem in the metric theory of continued fractions begins this way: Select a non-negative real function $f$ defined on the positive integers and a real number $x$, and form the partial sums $s_n$ of $f$ evaluated at the partial quotients $a_1,..., a_n$ in the continued fraction expansion for $x$. Does the sequence $\{s_n/n\}$ have a limit as $n\rar\infty$? In 1935 A. Y. Khinchin proved that the answer is yes for almost every $x$, provided that the function $f$ does not grow too quickly. In this paper we are going to explore a natural reformulation of this problem in which the function $f$ is defined on the rationals and the partial sums in question are over the intermediate convergents to $x$ with denominators less than a prescribed amount. By using some of Khinchin's ideas together with more modern results we are able to provide a quantitative asymptotic theorem analogous to the classical one mentioned above.
We investigate a collection of orthonormal functions that encodes information about the continued fraction expansion of real numbers. When suitably ordered these functions form a complete system of martingale differences and are a special case of a class of martingale differences considered by R. F. Gundy. By applying known results for martingales we obtain corresponding metric theorems for the continued fraction expansion of almost all real numbers.