A strengthening of the Bourgain-Kontorovich method: three new theorems

I. D. Kan

Sbornik. Mathematics, Tome 212 (2021) no. 7, pp. 921-964

Résumé

Consider the set $\mathfrak{D}_{\mathbf{A}}$ of irreducible denominators of the rational numbers representable by finite continued fractions all of whose partial quotients belong to some finite alphabet $\mathbf{A}$. Let the set of infinite continued fractions with partial quotients in this alphabet have Hausdorff dimension $\Delta_{\mathbf{A}}$ satisfying $\Delta_{\mathbf{A}} \geqslant0.7748\dots$ . Then $\mathfrak{D}_{\mathbf{A}}$ contains a positive share of positive integers. A previous similar result of the author of 2017 was related to the inequality $\Delta_{\mathbf{A}} >0.7807\dots$; in the original 2011 Bourgain-Kontorovich paper, $\Delta_{\mathbf{A}} >0.9839\dots$ . Bibliography: 28 titles.

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 intégerales multiples”, Applications of number theory to numerical analysis (Univ. Montreal, Montreal, QC, 1971), Academic Press, New York, 1972, 39–119 | MR | Zbl

[2] N. M. Korobov, Teoretiko-chislovye metody v priblizhennom analize, Fizmatgiz, M., 1963, 224 pp. | MR | Zbl

[3] H. Niederreiter, “Dyadic fractions with small partial quotients”, Monatsh. Math., 101:4 (1986), 309–315 | DOI | MR | Zbl

[4] 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 | Zbl

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

[6] N. G. Moshchevitin, On some open problems in Diophantine approximation, arXiv: 1202.4539

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

[8] D. A. Frolenkov, I. D. Kan, A reinforsment of the Bourgain–Kontorovich's theorem by elementary methods, arXiv: 1207.4546

[9] D. A. Frolenkov, I. D. Kan, A reinforsment of the Bourgain–Kontorovich's theorem, arXiv: 1207.5168

[10] I. D. Kan, D. A. Frolenkov, “A strengthening of a theorem of Bourgain and Kontorovich”, Izv. Math., 78:2 (2014), 293–353 | DOI | DOI | MR | Zbl

[11] 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 | Zbl

[12] Shinn Yih Huang, “An improvement to Zaremba's conjecture”, Geom. Funct. Anal., 25:3 (2015), 860–914 | DOI | MR | Zbl

[13] I. D. Kan, “A strengthening of a theorem of Bourgain and Kontorovich. III”, Izv. Math., 79:2 (2015), 288–310 | DOI | DOI | MR | Zbl

[14] I. D. Kan, “A strengthening of a theorem of Bourgain and Kontorovich. IV”, Izv. Math., 80:6 (2016), 1094–1117 | DOI | DOI | MR | Zbl

[15] I. D. Kan, “A strengthening of a theorem of Bourgain and Kontorovich. V”, Proc. Steklov Inst. Math., 296 (2017), 125–131 | DOI | DOI | MR | Zbl

[16] I. D. Kan, “Is Zaremba's conjecture true?”, Sb. Math., 210:3 (2019), 364–416 | DOI | DOI | MR | Zbl

[17] M. Magee, H. Oh, D. Winter, “Uniform congruence counting for Schottky semigroups in $\operatorname{SL}_2(\mathbf Z)$”, J. Reine Angew. Math., 2019:753 (2019), 89–135 ; arXiv: 1601.03705 | DOI | MR | Zbl

[18] I. D. Shkredov, Growth in Chevalley groups relatively to parabolic subgroups and some applications, arXiv: 2003.12785

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

[20] N. G. Moshchevitin, I. D. Shkredov, “On a modular form of Zaremba's conjecture”, Pacific J. Math., 309:1 (2020), 195–211 ; arXiv: 1911.07487 | DOI | MR | Zbl

[21] 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 | Zbl

[22] I. D. Kan, “Inversion of the Cauchy–Bunyakovskii–Schwarz inequality”, Math. Notes, 99:3 (2016), 378–381 | DOI | DOI | MR | Zbl

[23] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete mathematics. A foundation for computer science, 2nd ed., Addison-Wesley Publ. Co., Reading, MA, 1994, xiv+657 pp. | MR | Zbl

[24] 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 | DOI | MR | Zbl

[25] R. C. Vaughan, The Hardy–Littlewood method, Cambridge Tracts in Math., 80, Cambridge Univ. Press, Cambridge–New York, 1981, xi+172 pp. | MR | MR | Zbl | Zbl

[26] I. M. Vinogradov, “An estimate for a certain sum extended over the primes of an arithmetical progression”, Amer. Math. Soc. Transl. Ser. 2, 82, Amer. Math. Soc., Providence, RI, 1969, 147–164 | DOI | MR | Zbl

[27] S. V. Konyagin, “Otsenki trigonometricheskikh summ po podgruppam i summ Gaussa”, IV mezhdunarodnaya konferentsiya “Sovremennye problemy teorii chisel i ee prilozheniya”, posvyaschennaya 180-letiyu P. L. Chebysheva i 110-letiyu I. M. Vinogradova. Aktualnye problemy, Ch. 3 (Tula, 2001), MGU, mekh.-matem. fak-t, M., 2002, 86–114 | MR | Zbl

[28] N. M. Korobov, Exponential sums and their applications, Math. Appl. (Soviet Ser.), 80, Kluwer Acad. Publ., Dordrecht, 1992, xvi+209 pp. | DOI | MR | MR | Zbl | Zbl