Мақаланы E-mail арқылы жіберу Some applications of growth in $\mathrm{SL}_2(\pmb{\mathbb{F}}_p)$ to the proof of modular versions of Zaremba's conjecture
- Авторлар: Lyamkin M.V.1
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Мекемелер:
- Steklov Mathematical Institute of Russian Academy of Sciences
- Шығарылым: Том 213, № 10 (2022)
- Беттер: 108-129
- Бөлім: Articles
- URL: https://bakhtiniada.ru/0368-8666/article/view/133472
- DOI: https://doi.org/10.4213/sm9707
- ID: 133472
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Аннотация
Using growth in $\mathrm{SL}_2(\mathbb{F}_p)$ we prove that for every prime number $p$ and any positive integer $u$ there are positive integers $q=O(p^{2+\varepsilon})$, $\varepsilon > 0$, $q \equiv u \pmod{p}$, and $a < p$, $(a, p)=1$, such that the partial quotients of the continued fraction of $a/q$ are bounded by an absolute constant.Bibliography: 21 titles.
Негізгі сөздер
Авторлар туралы
Mikhail Lyamkin
Steklov Mathematical Institute of Russian Academy of Scienceswithout scientific degree, no status
Әдебиет тізімі
- J. Bourgain, A. Kontorovich, “On Zaremba's conjecture”, Ann. of Math. (2), 180:1 (2014), 137–196
- J. Bourgain, A. Gamburd, “Uniform expansion bounds for Cayley graphs of $operatorname{SL}_2(mathbb{F}_p)$”, Ann. of Math. (2), 167:2 (2008), 625–642
- L. E. Dickson, “Theory of linear groups in an arbitrary field”, Trans. Amer. Math. Soc., 2:4 (1901), 363–394
- L. E. Dickson, Linear groups: With an exposition of the Galois field theory, With an introduction by W. Magnus, Dover Publications, Inc., New York, 1958, xvi+312 pp.
- W. T. Gowers, “Quasirandom groups”, Combin. Probab. Comput., 17:3 (2008), 363–387
- H. A. Helfgott, “Growth and generation in $operatorname{SL}_2(mathbb{Z}/pmathbb{Z})$”, Ann. of Math. (2), 167:2 (2008), 601–623
- D. A. Hensley, “The distribution $operatorname{mod} n$ of fractions with bounded partial quotients”, Pacific J. Math, 166:1 (1994), 43–54
- D. Hensley, “The distribution of badly approximable rationals and continuants with bounded digits. II”, J. Number Theory, 34:3 (1990), 293–334
- D. Hensley, “The distribution of badly approximable numbers and continuants with bounded digits”, Theorie des nombres (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, 371–385
- А. Я. Хинчин, Цепные дроби, 3-е изд., Физматгиз, Л., 1961, 112 с.
- Р. Хорн, Ч. Джонсон, Матричный анализ, Мир, М., 1989, 656 с.
- O. Jenkinson, M. Pollicott, “Computing the dimension of dynamically defined sets: $E_2$ and bounded continued fractions”, Ergodic Theory Dynam. Systems, 21:5 (2001), 1429–1445
- A. Kontorovich, “Levels of distribution and the affine sieve”, Ann. Fac. Sci. Toulouse Math. (6), 23:5 (2014), 933–966
- E. Kowalski, “Explicit growth and expansion for $operatorname{SL}_2$”, Int. Math. Res. Not. IMRN, 2013:24 (2013), 5645–5708
- M. Magee, Hee Oh, D. Winter, “Uniform congruence counting for Schottky semigroups in $operatorname{SL}_2(mathbb{Z})$”, J. Reine Angew. Math., 2019:753 (2019), 89–135
- N. Moshchevitin, B. Murphy, I. Shkredov, “Popular products and continued fractions”, Israel J. Math., 238:2 (2020), 807–835
- N. G. Moshchevitin, I. D. Shkredov, “On a modular form of Zaremba's conjecture”, Pacific J. Math., 309:1 (2020), 195–211
- М. А. Наймарк, Теория представлений групп, Наука, М., 1976, 560 с.
- M. Rudnev, I. D. Shkredov, “On the growth rate in $operatorname{SL}_2(mathbb{F}_p)$, the affine group and sum-product type implications”, Mathematika, 68:3 (2022), 738–783
- I. D. Shkredov, Growth in Chevalley groups relatively to parabolic subgroups and some applications, 2020
- И. Д. Шкредов, “Некоммутативные методы в аддитивной комбинаторике и теории чисел”, УМН, 76:6(462) (2021), 119–180
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