CIRM - Videos & books Library - Zaremba's conjecture and growth in groups

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Authors : Shkredov, Ilya (Author of the conference)
CIRM (Publisher )

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Abstract : Zaremba's conjecture belongs to the area of continued fractions. It predicts that for any given positive integer q there is a positive a, a < q, (a,q)=1 such that all partial quotients b_j in its continued fractions expansion a/q = 1/b_1+1/b_2 +... + 1/b_s are bounded by five. At the moment the question is widely open although the area has a rich history of works by Korobov, Hensley, Niederreiter, Bourgain and many others. We survey certain results concerning this hypothesis and show how growth in groups helps to solve different relaxations of Zaremba's conjecture. In particular, we show that a deeper hypothesis of Hensley concerning some Cantor-type set with the Hausdorff dimension >1/2 takes place for the so-called modular form of Zaremba's conjecture.

Keywords : continued fractions; diophantine approximations; growth in groups; Chevalley groups

MSC Codes :
11A55 - Continued fractions, {For approximation results, See 11J70; See also 11K50, 30B70, 40A15}
11J70 - Continued fractions and generalizations
20G05 - Representation theory of linear algebraic groups
20G40 - Linear algebraic groups over finite fields
11B30 - Arithmetic combinatorics; higher degree uniformity

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Hennenfent, Guillaume

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https://videos.cirm-math.fr/2020-11-25 Shkredov.mp4

Information on the Event

Event Title : Jean-Morlet Chair 2020 - Conference: Diophantine Problems, Determinism and Randomness / Chaire Jean-Morlet 2020 - Conférence : Problèmes diophantiens, déterminisme et aléatoire
Event Organizers : Rivat, Joël ; Tichy, Robert
Dates : 23/11/2020 - 27/11/2020
Event Year : 2020
Event URL : https://www.chairejeanmorlet.com/2256.html

Citation Data

DOI : 10.24350/CIRM.V.19689203
Cite this video as: Shkredov, Ilya (2020). Zaremba's conjecture and growth in groups. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19689203
URI : http://dx.doi.org/10.24350/CIRM.V.19689203

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Bibliography

HENSLEY, Doug. Continued fraction Cantor sets, Hausdorff dimension, and functional analysis. Journal of number theory, 1992, vol. 40, no 3, p. 336-358. - https://doi.org/10.1016/0022-314X(92)90006-B

HELFGOTT, Harald Andrés. Growth and generation in $SL_2 (\mathbb{Z}/p\mathbb{Z})$. Annals of Mathematics, 2008, p. 601-623. - https://www.jstor.org/stable/40345357

KOROBOV, Nikolai Mikhailovich. Number-theoretic methods in approximate analysis. 1963. -

SARNAK, Peter, XUE, Xiaoxi, et al. Bounds for multiplicities of automorphic representations. Duke Mathematical Journal, 1991, vol. 64, no 1, p. 207-227. - http://dx.doi.org/10.1215/S0012-7094-91-06410-0

ZAREMBA, Stanisłlaw K. La méthode des “bons treillis” pour le calcul des intégrales multiples. In : Applications of number theory to numerical analysis. Academic Press, 1972. p. 39-119. - https://doi.org/10.1016/B978-0-12-775950-0.50009-1

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