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 Farey fractions with denominators restricted to lie in arithmetic progression.
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We prove some asymptotic results on the distribution of h-tuples of consecutive Farey fractions of order Q with odd denominators as $Q\to \infty$.
Pair correlation of Farey fractions with square-free denominators
本文研究分母为无平方因子数的 Farey 分数的对关联问题，证明其极限对关联函数存在并给出显式公式，推进了数论中 Farey 分数分布性质的相关研究。
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].
Let $F_Q$ be the set of Farey fractions of order $Q$. Given the integers $\d\ge 2$ and $0\le \c \le \d-1$, let $F_Q(c,d)$ be the subset of $F_Q$ of those fractions whose denominators are $\equiv c \pmod d$, arranged in ascending order. The problem we address here is to show that as $Q\to\infty$, there exists a limit probability measuring the distribution of $s$-tuples of consecutive denominators of fractions in $F_Q(c,d)$. This shows that the clusters of points $(q_0/Q,q_1/Q,...,q_s/Q)\in[0,1]^{s+1}$, where $q_0,q_1,...,q_s$ are consecutive denominators of members of $F_Q$ produce a limit set,…
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.
Denominators in the Farey Sequence, the Stern Sequence, and the Frobenius Problem with Relatively Prime Multipliers
该研究针对带系数互素限制的弗罗贝尼乌斯开放问题，证明此类问题仍存在无法表示的最大整数，并探究其与斯特恩序列等的关联及相关性质。
In this article, we study the distribution of index of Farey fractions which was first introduced and studied by Hall and Shiu. We provide asymptotic formulas for moments of index of Farey fractions twisted by Dirichlet characters for Farey fractions with $\mathcal{B}$-free denominators. Additionally, we reconsider the squarefree case earlier done in [ALVZ08], and obtain new results for moments of indices with square-free denominators. We also study higher level correlations of the index function generalizing earlier known results on two level correlations.
On denominators of consecutive $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions
研究连续SL(2,N)-饱和 Farey 分数的分母在特定区域的稠密性，并给出其分布公式。
The pair correlations of Farey fractions with denominators $q$ satisfying $(q,m)=1$, respectively $q\equiv b \pmod{m}$ with $(b,m)=1$, are shown to exist and are explicitly computed.
Let $F_Q$ be the Farey sequence of order $Q$ and let $F_{Q,o}$ and $F_{Q,e}$ be the set of those Farey fractions of order $Q$ with odd, respectively even denominators. A fundamental property of $F_Q$ says that the sum of denominators of any pair of neighbor fractions is always greater than $Q$. This property fails for $F_{Q,o}$ and for $F_{Q,e}$. The local density, as $Q\to\infty$, of the normalized pairs $(q'/Q,q''/Q)$, where $q',q''$ are denominators of consecutive fractions in $F_{Q,o}$, was computed previously. The density increases over a series of quadrilateral steps ascending in a harmo…
Distribution of Farey fractions with $k$-free denominators
该研究探讨带k自由分母的Farey分数序列的分布与局部统计性质，证明其模1均匀分布，推导广义黎曼假设的等价形式，建立高阶相关测度公式，为相关数论问题提供新结果。
The denominators of power sums of arithmetic progressions
该研究将幂和多项式分母的相关结果推广到算术级数幂和，得到系数整性的显式判据，并应用于伯努利多项式分母序列，刻画了相邻分母商中无穷多出现的整数。
On the distribution of polynomial Farey points and Chebyshev's bias phenomenon
该文研究多项式Farey分数的分布与Chebyshev偏差问题，通过偏差和配对关联测度刻画其全局与局部分布，证明均匀分布性，给出特定多项式的配对关联函数显式公式，并得到计数函数误差项的Ω结果。
The {Farey fractions} $F_n$ of order $n$ consist of all fractions $\frac{h}{k}$ in lowest terms lying in the closed unit interval and having denominator at most $n$. This paper considers the products $F_n$ of all nonzero Farey fractions of order $n$. It studies their growth measured by $\log(F_n)$ and their divisibility properties by powers of a fixed prime, given by $ord_p(F_n)$, as a function of $n$. The growth of $\log(F_n)$ is related to the Riemann hypothesis. This paper theoretically and empirically studies the functions $ord_p(F_n)$ and formulates several unproved properties (P1)-(P4) t…
The Farey sequence is the sequence of all rational numbers in the real unit interval, stratified by increasing denominators. A classical result by Hall says that its normalized gap distribution is the same as the distribution of the random variable 1/(2 zeta(2) xy) on a certain unit triangle. In this paper we weight the denominators by an arbitrary piecewise-smooth continuous function, and we characterize the resulting gap distribution as that of a multiple of the above variable, defined on a certain unit pentagon. Our characterization refines previous results by Boca, Cobeli and Zaharescu, bu…
Smallest denominators
该论文将Chen-Haynes定理及Kruwswijk-Meijer猜想推广至高维，用多维Farey空隙统计刻画最小分母分布，证明分布函数与有限矩收敛，并推广Kargaev-Zhigljavsky矩结果，获得鸽笼统计新结果。
There are two questions in analytic number theory which have attracted much attention over the years. The first one is about the asymptotic formula for the variance associated with the distribution of a real sequence in arithmetic progressions, which origins in the work of Barban. The second one is about the function $r_3(n)$, the number of ordered representation of $n$ in the sum of three positive cubes, and several results are recently proved by Robert C. Vaughan. The paper is concerned with the conjunction of these questions, with the application of the Hardy-Littlewood Method and the Farey…
A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a metrical and a non-metrical point of view and, on the other hand, from an asymptotic and also a uniform point of view. The principal novelty is a Khintchine type theorem for uniform approximation in this context. Some applications of this theory are also discussed.
A distribution related to Farey sequences -- I
本文采用Boca等人引入的方法研究Farey序列的算术性质，针对特定分母同余的分数进行染色并分析间隙分布，最终推导出D=2,3且c=0时极限比例ν(r;D,c)的显式公式。
A distribution related to Farey sequences -- II
本文沿用 Boca、Cobeli 和 Zaharescu 的方法研究 Farey 序列间隔分布，针对模 3 分母染色情形给出极限比例 ν(r;3,c) 的显式公式。
Rational dynamics of a prime-representing map
本文研究了素数表示常数递归构造中出现的映射$\mathcal{T}(x)=\lfloor x floor(1+\{x\})$的有理动力学，定义了有理数的阶，并证明了有限阶有理数的自然密度为1，同时给出了特定分母下的显式分数家族。
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