Modular Generalization of the Bourgain--Kontorovich Theorem

I. D. Kan

Matematičeskie zametki, Tome 114 (2023) no. 5, pp. 739-752.

Résumé

The set $\mathfrak{D}^N_\mathbf{A}$ of all irreducible denominators $\le N$ of positive rationals $1$ whose continued fraction expansions consist of elements of the set $\mathbf{A}=\{1,2,4\}$ is considered. We prove that, for any prime $Q\le N^{2/3}$, the set $\mathfrak{D}^N_{\mathbf{A}}$ contains almost all possible remainders on division by $Q$ and the remainder term in the corresponding asymptotic formula decays according to a power law.

Détail

Keywords: continued fraction, trigonometric sum
Mots-clés : Zaremba's conjecture, Hausdorff dimension.

Bibliographie

[1] S. K. Zaremba, “La méthode des “bons treillis” pour le calcul des intégrales multiples”, Applications of Number Theory to Numerical Analysis (Montreal, Canada, 1971), Academic Press, New York, 1972, 39–119 | MR

[2] J. Bourgain, A. Kontorovich, “On Zaremba's conjecture”, Ann. of Math. (2), 180:1 (2014), 137–196 | DOI | MR

[3] N. G. Moshchevitin, On Some Open Problems in Diophantine Approximation, arXiv: 1202.4539

[4] N. M. Korobov, Teoretiko-chislovye metody v priblizhennom analize, Fizmatgiz, M., 1963 | MR

[5] N. M. Korobov, “Vychislenie kratnykh integralov metodom optimalnykh koeffitsientov”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 1959, no. 4, 19–25 | MR

[6] I. D. Kan, “Usilenie odnoi teoremy Burgeina–Kontorovicha”, Dalnevost. matem. zhurn., 20:2 (2020), 164–190 | DOI

[7] I. D. Kan, “Usilenie metoda Burgeina–Kontorovicha: tri novykh teoremy”, Matem. sb., 212:7 (2021), 39–83 | DOI

[8] J. Bourgain, A. Kontorovich, “On Zaremba's conjecture”, C. R. Math. Acad. Sci. Paris, 349:9–10 (2011), 493-495 | MR

[9] D. Hensley, “The Hausdorff dimensions of some continued fraction Cantor sets”, J. Number Theory, 33:2 (1989), 182–198 | DOI | MR

[10] D. A. Frolenkov, I. D. Kan, A Reinforsment of the Bourgain–Kontorovich's Theorem by Elementary Methods, arXiv: 1207.4546

[11] D. A. Frolenkov, I. D. Kan, A Reinforsment of the Bourgain–Kontorovich's Theorem, arXiv: 1207.5168

[12] I. D. Kan, D. A. Frolenkov, “Usilenie teoremy Burgeina–Kontorovicha”, Izv. RAN. Ser. matem., 78:2 (2014), 87–144 | DOI | MR | Zbl

[13] D. A. Frolenkov, I. D. Kan, “A strengthening of a theorem of Bourgain–Kontorovich. II”, Mosc. J. Comb. Number Theory, 4:1 (2014), 78–117 | MR

[14] I. D. Kan, “Usilenie teoremy Burgeina–Kontorovicha. III”, Izv. RAN. Ser. matem., 79:2 (2015), 77–100 | DOI | MR | Zbl

[15] I. D. Kan, “Usilenie teoremy Burgeina–Kontorovicha. IV”, Izv. RAN. Ser. matem., 80:6 (2016), 103–126 | DOI | MR

[16] I. D. Kan, “Usilenie teoremy Burgeina–Kontorovicha. V”, Analiticheskaya i kombinatornaya teoriya chisel, Trudy MIAN, 296, Nauka, M., 2017, 133–139 | DOI | MR

[17] I. D. Kan, “Verna li gipoteza Zaremby?”, Matem. sb., 210:3 (2019), 75–130 | DOI | MR

[18] I. D. Kan, “Usilenie teoremy Burgeina–Kontorovicha o malykh znacheniyakh khausdorfovoi razmernosti”, Funkts. analiz i ego pril., 56:1 (2022), 66–80 | DOI

[19] I. D. Kan, “Usilenie metoda Burgeina–Kontorovicha: tri novykh teoremy”, Matem. sb., 212:7 (2021), 39–83 | DOI

[20] M. V. Lyamkin, “O prilozheniyakh rosta v $\mathrm{SL}_2(\mathbb{F}_p)$ k dokazatelstvu modulyarnykh variantov gipotezy Zaremby”, Matem. sb., 213:10 (2022), 108–129 | DOI | MR

[21] I. D. Shkredov, “Growth in Chevalley groups relatively to parabolic subgroups and some applications”, Rev. Mat. Iberoam., 38:6 (2022), 1945–1973 | DOI | MR

[22] N. G. Moshchevitin, B. Murphy, I. Shkredov, “Popular products and continued fractions”, Israel J. Math., 238:2 (2020), 807–835 | MR

[23] N. G. Moshchevitin, I. D. Shkredov, “On a modular form of Zaremba's conjecture”, Pacific J. Math., 309:1 (2020), 195–211 | DOI | MR

[24] M. Magee, H. Oh, D. Winter, “Uniform congruence counting for Schottky semigroups in $\mathrm{SL}_2(\mathbb Z)$”, J. Reine Angew. Math., 753 (2019), 89–135 | DOI | MR

[25] O. Jenkinson, “On the density of Hausdorff dimensions of bounded type continued fraction sets: the Texan conjecture”, Stoch. Dyn., 4:1 (2004), 63–76 | DOI | MR

[26] M. Pollicott, P. Vytnova, “Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups”, Trans. Amer. Math. Soc. Ser. B, 9 (2022), 1102–1159 | MR

[27] R. Graham, D. Knuth, O. Patashnik, Concrete Mathematics. A Foundation for Computer Science, Addison-Wesley, Reading, MA, 1994 | MR

[28] D. Hensley, “A polynomial time algorithm for the Hausdorff dimension of continued fraction Cantor sets”, J. Number Theory, 58:1 (1996), 9–45 | DOI | MR

[29] D. Hensley, “The distribution of badly approximable numbers and continuants with bounded digits”, Théorie des nombres (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, 371–385 | MR

[30] N. M. Korobov, Trigonometricheskie summy i ikh prilozheniya, GIFML, M., 1989 | MR

[31] S. V. Konyagin, “Otsenki trigonometricheskikh summ po podgruppam i summ Gaussa”, IV Mezhdunarodnaya konferentsiya “Sovremennye problemy teorii chisel i ee prilozheniya”, ch. III (Tula, 2001), Izd-vo Mosk. un-ta, M., 2002, 86–114 | MR

[32] I. D. Kan, “Obraschenie neravenstva Koshi–Bunyakovskogo–Shvartsa”, Matem. zametki, 99:3 (2016), 361–365 | DOI | MR

[33] I. D. Kan, “Lineinye sravneniya v tsepnykh drobyakh iz konechnykh alfavitov”, Matem. zametki, 103:6 (2018), 853–862 | DOI | MR