A distribution related to Farey series

M. A. Korolev

Čebyševskij sbornik, Tome 24 (2023) no. 4, pp. 137-190.

Résumé

We study some arithmetical properties of Farey fractions by the method introduced by F. Boca, C. Cobeli and A. Zaharescu (2001). Suppose that $D\geqslant 2$ is a fixed integer and denote by $\Phi_{Q}$ the classical Farey series of order $Q$. Now let us colour to the red the fractions in $\Phi_{Q}$ with denominators divisible by $D$. Consider the gaps in $\Phi_{Q}$ with coloured endpoints, that do not contain the fractions $a/q$ with $D|q$ inside. The question is to find the limit proportions $\nu(r;D)$ (as $Q\to +\infty$) of such gaps with precisely $r$ fractions inside in the whole set of the gaps under considering ($r = 1,2,3,\ldots$). In fact, the expression for this proportion can be derived from the general result obtained by C. Cobeli, M. Vâjâitu and A. Zaharescu (2014). However, such formula expresses $\nu(r;D)$ in the terms of areas of some polygons related to some geometrical transform of «Farey triangle», that is, the subdomain of unit square defined by $x+y>1$, $0$. In the present paper, we obtain the precise formulas for $\nu(r;D)$ (in terms of the parameter $r$, $r=1,2,3,\ldots$) for the cases $D = 2, 3$.

Détail

Keywords: Farey series, Farey fractions, Farey triangle, arithmetical properties, distribution, $BCZ$-transform.

Bibliographie

[1] Cobeli C., Zaharescu A., “The Haros-Farey sequence at two hundred years. A survey”, Acta Univ. Apulensis Math. Inform., 5 (2003), 1–38 | MR | Zbl

[2] Haynes A., “A note on Farey fractions with odd denominators”, J. Number Theory, 98:2 (2003), 89–104 | DOI | MR | Zbl

[3] Boca F.P., Cobeli C., Zaharescu A., “On the distribution of the Farey sequence with odd denominators”, Michigan Math. J., 51 (2003), 557–573 | DOI | MR | Zbl

[4] Cobeli C., Zaharescu A., The distribution of rationals in residue classes, 14.11.2005, arXiv: math/0511356v1 [math.NT]

[5] Cobeli C., Zaharescu A., “On the Farey Fractions with Denominators in Arithmetic Progression”, J. Integer Sequences, 9 (2006), 06.3.4 | MR | Zbl

[6] Alkan E., Ledoan A.H., Vâjâitu M., Zaharescu A., “Discrepancy of Fractions with Divisibility Constraints”, Monatsh. Math., 149 (2006), 179–192 | DOI | MR | Zbl

[7] Haynes A., The distribution of special subsets of the Farey sequence, 13.07.2009, arXiv: math/0907.2171v1 [math.NT]

[8] Cobeli C., Vâjâitu M., Zaharescu A., “The distribution of rationals in residue classes”, Math. Reports, 14:1(64) (2012), 1–19 | MR | Zbl

[9] Badziahin D.A., Haynes A.K., “A note on Farey fractions with denominators in arithmetic progressions”, Acta Arith., 146:3 (2011), 205–215 | DOI | MR

[10] Boca F.P., Heersink B., Spiegelhalter P., Gap distribution of Farey fractions under some divisibility constraints, 2013, arXiv: 1301.0277v2 [math.NT] | MR

[11] Bittua, Chaubeya S., Goela S., On the distribution of index of Farey Sequences, 2022, arXiv: 2203.16215v1 [math.NT] | MR

[12] Boca F.P., Siskaki M., A note on the pair correlation of Farey fractions, 19.09.2022, arXiv: 2109.12744v2 [math.NT] | MR

[13] Boca F.P., Cobeli C., Zaharescu A., “A conjecture of R. R. Hall on Farey points”, J. reine angew. Math., 535 (2001), 207–236 | MR | Zbl

[14] Baker A., Transcendental number theory, Cambr. Univ. Press, 1975 | MR | Zbl

[15] Lindeman F., “Ueber die Zahl $\pi$”, Math. Ann., 20 (1882), 213–225 | DOI | MR

[16] Ustinov A.V., “The solution of Arnold's problem on the weak asymptotics of Frobenius numbers with three arguments”, Sb. Math., 200:4 (2009), 597–627 | DOI | DOI | MR | Zbl

[17] Graham R.L., Knuth D.E., Patashnik O, Concrete Mathematics, 2nd ed., Addison-Wesley, Massachusetts, 1994 | MR | MR | Zbl

[18] Smirnov E.Yu., Frieze patterns and continued fractions, MCCME, M., 2022 (in Russian)

[19] Boca F.P., Gologan R., Zaharescu A., “On the index of Farey sequences”, Quart. J. Math., 53:4 (2002), 377–391 | DOI | MR | Zbl

[20] Postnikov A.G., Introduction to analytic number theory, Transl. Math. Monographs, 68, AMS, Providence, Rhode Island, 1988 | DOI | MR | Zbl

[21] Gradstein I.S., Ryjik I.M., Tables of Integrals, Sums and Products, Academic Press, New York, 1965 | MR | MR