\(k\)-moments of distances between centers of Ford circles
type: Item
instance of: scholarly article
title: \(k\)-moments of distances between centers of Ford circles
zbMATH Open document ID: 1308.11017
author: Sneha Chaubey
author: Amita Malik
author: Alexandru Zaharescu
published in: Journal of Mathematical Analysis and Applications
publication date: 2014-10-31T00:00:00Z
reviewed by: Dmitriy Frolenkov
Mathematics Subject Classification ID: 11B57
Mathematics Subject Classification ID: 51M99
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11L05
zbMATH DE Number: 6362854
zbMATH Keywords: Ford circles
zbMATH Keywords: Farey fractions
zbMATH Keywords: Riemann zeta function
zbMATH Keywords: Kloosterman sums
MaRDI profile type: Publication
full work available at URL: j.jmaa.2014.09.007
OpenAlex ID: W1970809463
cites work: A Poincaré section for the horocycle flow on the space of lattices
cites work: A note on Farey fractions with denominators in arithmetic progressions
cites work: A conjecture of R. R. Hall on Farey points
cites work: THE CORRELATIONS OF FAREY FRACTIONS
cites work: Q3116334
cites work: Q5492392
cites work: On Kloosterman's sum
cites work: Fractions
cites work: A note on Farey fractions with odd denominators.
cites work: An Asymptotic Formula in the Theory of Numbers
cites work: Q3257210
cites work: Q3123961
cites work: Q3735790
cites work: Q3276744
cites work: On Some Exponential Sums
DOI: 10.1016/J.JMAA.2014.09.007
Recommended article: Higher moments of distances between consecutive Ford spheres
Recommended article: First moment of distances between centres of Ford spheres
Recommended article: Über Gleichverteilung bei FORD-Kreisen
Recommended article: Limiting distribution of the maximal distance between random points on a circle: a moments approach
Recommended article: The center of distances of central Cantor sets
Recommended article: Q2979225
Recommended article: Distribution and generalized centre in planar nearrings
Recommended article: On the number of points with pairwise integral distances on a circle
Recommended article: Center of distances and central Cantor sets
Recommended article: Q4263843
review text: The subject matter of the paper is the investigation of the distribution of Ford circles. A Ford circle is a circle with center at \((p/q,1/(2q^2))\) and radius \(1/(2q^2),\) where \(p,q\) are coprime positive integers. The authors consider the set \(\mathcal{F}_{I,Q}\) of the Ford circles being tangent to the real axis at points \((a/q,0)\) where \(\frac{a}{q}\in I=[\alpha,\beta]\subseteq[0,1], \; q\leq Q.\) Let \(N_I(Q)=\#\mathcal{F}_{I,Q},\) and let NEWLINE\[NEWLINEO_{Q,j},\quad j=1,\ldots,N_I(Q)NEWLINE\]NEWLINE be centers of the circles. Define the following moments NEWLINE\[NEWLINE \mathcal{M}_{k,I}(Q)=\frac{1}{|I|}\sum_{j=1}^{N_I(Q)-1}\left(\text{dist}(O_{Q,j},O_{Q,j+1})\right)^k,NEWLINE\]NEWLINE NEWLINE\[NEWLINE \mathcal{A}_{k,I}(X)=\frac{1}{X}\int_X^{2X}\mathcal{M}_{k,I}([Y])\,dY. NEWLINE\]NEWLINE In the paper for any positive integer \(k\) very precise asymptotic formulas for \(\mathcal{M}_{k,I}(Q), \mathcal{A}_{k,I}(X)\) are obtained. The proof is based on the bijection of \(\mathcal{F}_{I,Q}\) with the set of Farey fractions \(\mathcal{F}_{Q}.\) This allows to rewrite the moments in terms of Farey fractions and different moments of Farey fractions. Using the properties of Farey fractions and Perron's formula the problem is reduced to evaluation of integrals involving the Riemann zeta function. To estimate the integrals the estimates on zeta function (also in the Vinogradov-Korobov zero free region) are used. To obtain asymptotic formulas for desired moment of Farey fractions Weil bounds for Kloosterman sums were used see \textit{F. P. Boca} et al. [J. Reine Angew. Math. 535, 207--236 (2001; Zbl 1006.11053)].
description: scientific article; zbMATH DE number 6362854