chebЧебышевский сборникChebyshevskii Sbornik2226-8383Tula State Lev Tolstoy Pedagogical University10.22405/2226-8383-2025-26-3-136-173cheb-2011Research ArticleСтатьиArticleОб одном распределении, связанном с рядами Фарея - IIA distribution related to Farey series - IIКоролёвМаксим АлександровичKorolevMaxim Alexandrovichkorolevma@mi-ras.ruМатематический институт им. В. А. Стеклова Российской Академии наукРоссияSteklov Mathematical Institute of Russian Academy of SciencesRussian Federation202501112025263136173Copyright © Королёв М.А., 20252025Королёв М.А.Korolev M.A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.chebsbornik.ru/jour/article/view/2011

В настоящей статье продолжены исследования некоторых арифметических свойств рядов Фарея с помощью метода, созданного Ф. Бока, К. Кобели и А. Захареску (2001).Пусть Φ𝑄 — классический ряд Фарея порядка 𝑄. Задавшись фиксированными числами 𝐷 ⩾ 2 и 0 ⩽ 𝑐0 ⩽ 𝐷 − 1, пометим красным цветом все дроби в Φ𝑄, знаменатели которых ≡ 𝑐0 (mod𝐷). Рассмотрим далее промежутки в Φ𝑄 с окрашенными концами, внутри которых нет других окрашенных дробей, т. е. дробей 𝑎/𝑞 с условием 𝑞 ≡ 𝑐0 (mod𝐷). Какова предельная (при 𝑄 → +∞) доля 𝜈(𝑟;𝐷, 𝑐0) промежутков, внутри которых имеется вточности 𝑟 неокрашенных дробей, в общем числе таких промежутков (𝑟 = 0, 1, 2, 3, . . .)?По сути, выражение для такой доли может быть получено из общих результатов, принадлежащих К. Кобели, М. Выжийту и А. Захареску (2012). Однако такое выражение для 𝜈(𝑟;𝐷, 𝑐0) представляет собой сумму площадей некоторых многоугольников, определяемых посредством специального геометрического преобразования. В настоящей статье мы получаем явные выражения для таких долей 𝜈(𝑟;𝐷, 𝑐0) для случаев 𝐷 = 3 и 𝑐0 = 1, 2. Тем самым с учётом предыдущей работы автора (2023) случай разности 𝐷 = 3 оказывается изученным полностью.

We continue to study some arithmetical properties of Farey sequences by the method introduced by F. Boca, C. Cobeli and A. Zaharescu (2001). Let Φ𝑄 be the classical Farey sequence of order 𝑄. Having the fixed integers 𝐷 ⩾ 2 and 0 ⩽ 𝑐0 ⩽ 𝐷 − 1, we colour to the redthe fractions in Φ𝑄 with denominators ≡ 𝑐0 (mod𝐷). Consider the gaps in Φ𝑄 with coloured endpoints, that do not contain the fractions 𝑎/𝑞 with 𝑞 ≡ 𝑐0 (mod𝐷) inside. The question is to find the limit proportions 𝜈(𝑟;𝐷, 𝑐0) (as 𝑄 → +∞) of such gaps with precisely 𝑟 fractions inside in the whole set of the gaps under considering (𝑟 = 0, 1, 2, 3, . . .).In fact, the expression for this proportion can be derived from the general result obtained by C. Cobeli, M. Vˆajˆaitu and A. Zaharescu (2012). However, such formula expresses 𝜈(𝑟;𝐷, 𝑐0) in the terms of areas of some polygons related to a special geometrical transform. In the present paper, we obtain explicit formulas for 𝜈(𝑟;𝐷, 𝑐0) for the cases 𝐷 = 3 and 𝑐0 = 1, 2. Thus this and previous author’s papers cover the case 𝐷 = 3.

ряд Фареядроби Фареятреугольник Фареяарифметические свойствараспределениеBCZ-преобразованиепреобразование Фарея.Farey seriesFarey fractionsFarey trianglearithmetical propertiesdistribution𝐵𝐶𝑍-transformFarey transform.Работа выполнена в МЦМУ МИАН при финансовой поддержке Минобрнауки России (соглашение № 075-15-2025-303).ReferencesКоролёв М.А., Об одном распределении, связанном с рядами Фарея // Чебышевский сб., 2023. Т. 24. № 4. С. 137–190.Korolev M.A., 2023, “A distribution related to Farey sequences”, Chebyshevskii Sb., vol. 24. № 4. P. 137–190.Korolev M.A., A distribution related to Farey sequences - I // arXiv:2502.19881 [math.NT] (27.02.2025).Korolev M.A., 2025, “A distribution related to Farey sequences - I”, arXiv:2502.19881 [math.NT] (27.02.2025).Gutthery S.B., A Motif of Mathematics // Docent Press, Boston, Massachusetts, USA. 2011.Gutthery S.B., 2011, “A Motif of Mathematics”, Docent Press, Boston, Massachusetts, USA.Cobeli C., Zaharescu A., The Haros-Farey sequence at two hundred years. A survey // Acta Univ. Apulensis Math. Inform. 2003. Vol. 5. P. 1–38.Cobeli C., Zaharescu A., 2003, “The Haros-Farey sequence at two hundred years. A survey”, Acta Univ. Apulensis Math. Inform., vol. 5. P. 1–38.Alkan E., Ledoan A.H., Vˆajˆaitu M., Zaharescu A., Discrepancy of fractions with divisibility constraints // Monatsh. Math. 2006. Vol. 149. P. 179–192.Alkan E., Ledoan A.H., Vˆajˆaitu M., Zaharescu A., 2006, “Discrepancy of fractions with divisibility constraints”, Monatsh. Math., vol. 149. P. 179–192.Alkan E., Ledoan A.H., Vˆajˆaitu M., Zaharescu A., Discrepancy of sets of fractions with congruence constraints // Rev. Roumaine Math. Pures Appl. 2006. Vol. 51. P. 265–276.Alkan E., Ledoan A.H., Vˆajˆaitu M., Zaharescu A., 2006, “Discrepancy of sets of fractions with congruence constraints”, Rev. Roumaine Math. Pures Appl., vol. 51. P. 265–276.Boca F.P., Heersink B., Spiegelhalter P., Gap distribution of Farey fractions under some divisibility constraints // Integers. 2013. Vol. 13. Paper No. A44. 15 pp.Boca F.P., Heersink B., Spiegelhalter P., 2013, “Gap distribution of Farey fractions under some divisibility constraints”, Integers, vol. 13. Paper No. A44. 15 pp.Heersink B., Poincare sections for the horocycle ow in covers of 𝑆𝐿(2;R) = 𝑆𝐿(2;Z) and applications to Farey fraction statistics // Monatsh. Math. 2016. Vol. 179. P. 389–420.Heersink B., 2016, “Poincare sections for the horocycle ow in covers of 𝑆𝐿(2;R) = 𝑆𝐿(2; Z) and applications to Farey fraction statistics”, Monatsh. Math., vol. 179. P. 389–420.Ledoan A.H., The discrepancy of Farey series // Acta Math. Hungar. 2018. Vol. 156. P. 465–480.Ledoan A.H., The discrepancy of Farey series // Acta Math. Hungar. 2018. Vol. 156. P. 465–480.Athreya J.S., Cheung Y., A Poincar´e section for horocycle flow on the space of lattices // arXiv:1206.6597 [math.DS] (28.06.2012).Athreya J.S., Cheung Y., 2012, “A Poincar´e section for horocycle flow on the space of lattices”, arXiv:1206.6597 [math.DS] (28.06.2012).Athreya J.S., Chaika J., Leli`evre S., The gap distribution of slopes on the golden 𝐿 // arXiv:1308.4203 [math.DS] (20.08.2013).Athreya J.S., Chaika J., Leli`evre S.,2013, “The gap distribution of slopes on the golden 𝐿”, arXiv:1308.4203 [math.DS] (20.08.2013).Taha D., The Boca-Cobeli-Zaharescu Map Analogue for the Hecke Triangle Groups 𝐺𝑞 // arXiv:1810.10668 [math.DS] (25.10.2018).Taha D., 2018, “The Boca-Cobeli-Zaharescu Map Analogue for the Hecke Triangle Groups 𝐺𝑞”, arXiv:1810.10668 [math.DS] (25.10.2018).Cheung Y., Quas A., BCZ map is weakly mixing // arXiv:2403.14976v1 [math.DS] (22.03.2024).Cheung Y., Quas A., 2024, “BCZ map is weakly mixing”, arXiv:2403.14976v1 [math.DS] (22.03.2024).Cobeli C., Zaharescu A., On the intervals of a third between Farey fractions // arXiv:math/0511363v1 [math.NT] (14.11.2005).Cobeli C., Zaharescu A., 2005, “On the intervals of a third between Farey fractions”, arXiv:math/0511363v1 [math.NT] (14.11.2005).Haynes A., The distribution of special subsets of the Farey sequence // J. Number Theory. 2004. Vol. 107. P. 95–104.Haynes A., 2004, “The distribution of special subsets of the Farey sequence”, J. Number Theory., vol. 107. P. 95–104.Haynes A.K., Numerators of differences of nonconsecutive Farey fractions // Int. J. NumberTheory. 2010. Vol. 6. № 3. P. 655–666.Haynes A.K., 2010, “Numerators of differences of nonconsecutive Farey fractions”, Int. J. Number Theory, vol. 6. № 3. P. 655–666.Boca F.P., Siskaki M., A note on the pair correlation of Farey fractions // arXiv:2109.12744v2 [math.NT] (19.09.2022).Boca F.P., Siskaki M., 2022, “A note on the pair correlation of Farey fractions”, arXiv: 2109.12744v2 [math.NT] (19.09.2022).Marklof J., Smallest denominators // arXiv:2310.11251v1 [math.NT] (17.10.2023).Marklof J., 2023, “Smallest denominators”, arXiv:2310.11251v1 [math.NT] (17.10.2023).Chahala B., Chaube S., Pair correlation of Farey fractions with square-free denominators // arXiv:2303.12882v1 [math.NT] (22.05.2023).Chahala B., Chaube S., 2023, “Pair correlation of Farey fractions with square-free denominators”, arXiv:2303.12882v1 [math.NT] (22.05.2023).Chahal B., Chaubey S., Goel S., On the distribution of index of Farey sequences // Res. Number Theory. 2024. Vol. 10. Article No 27. 34 pp.Chahal B., Chaubey S., Goel S., 2024, “On the distribution of index of Farey sequences”, Res. Number Theory., vol. 10. Article No 27. 34 pp.Chen H., Haynes A., Expected value of the smallest denominator in a random interval of fixed radius // arXiv:2109.12668v2 [math.NT] (15.11.2022).Chen H., Haynes A., 2024, “Expected value of the smallest denominator in a random interval of fixed radius”, arXiv:2109.12668v2 [math.NT] (15.11.2022).Sayous R., Gaps in the complex Farey sequence of an imaginary quadratic number field // arXiv:2407.04380v1 [math.NT] (05.07.2024).Sayous R., 2024, “Gaps in the complex Farey sequence of an imaginary quadratic number field”, arXiv:2407.04380v1 [math.NT] (05.07.2024).Ren X., On 𝑞-deformed Farey sum and a homological interpretation of 𝑞-deformed real quadratic irrational numbers // arXiv:2210.06056v4 [math.RT] (05.01.2024).Ren X., 2024, “On 𝑞-deformed Farey sum and a homological interpretation of 𝑞-deformed real quadratic irrational numbers”, arXiv:2210.06056v4 [math.RT] (05.01.2024).Boca F.P., Cobeli C., Zaharescu A., A conjecture of R. R. Hall on Farey points // J. reine angew. Math. 2001. Vol. 535. P. 207–236.Boca F.P., Cobeli C., Zaharescu A., 2001, “A conjecture of R. R. Hall on Farey points”, J. reine angew. Math., vol. 535. P. 207–236.Cobeli C., Zaharescu A., The distribution of rationals in residue classes // arXiv:math/0511356v1 [math.NT] (14.11.2005).Cobeli C., Zaharescu A., 2005, “The distribution of rationals in residue classes”, arXiv: math/0511356v1 [math.NT] (14.11.2005).Cobeli C., Vˆajˆaitu M., Zaharescu A., The distribution of rationals in residue classes // Math. Reports. 2012. Vol. 14(64). № 1. P. 1–19.Cobeli C., Vˆajˆaitu M., Zaharescu A.,2012, “The distribution of rationals in residue classes”, Math. Reports., vol. 14(64). № 1. P. 1–19.Boca F.P., Cobeli C., Zaharescu A., On the distribution of the Farey sequence with odd denominators // Michigan Math. J. 2003. Vol. 51. P. 557–573.Boca F.P., Cobeli C., Zaharescu A., 2003, “On the distribution of the Farey sequence with odd denominators”, Michigan Math. J., vol. 51. P. 557–573.Badziahin D.A., Haynes A.K., A note on Farey fractions with denominators in arithmetic progressions // Acta Arith., 2011. Vol. 146. № 3. P. 205–215.Badziahin D.A., Haynes A.K., 2011, “A note on Farey fractions with denominators in arithmetic progressions”, Acta Arith., vol. 146. № 3. P. 205–215.Смирнов Е.Ю., Фризы и цепные дроби. М., МЦНМО, 2022.Smirnov E.Yu., 2022, Frieze patterns and continued fractions (in Russian), M., MCCME, 2022.

The authors declare that there are no conflicts of interest present.