CIRM - Videos & books Library - Bounded remainder sets for rotations on $p$-adic solenoids

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Auteurs : Haynes, Alan (Auteur de la conférence)
CIRM (Editeur )

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bounded remainder sets Diophantine approximation rotations on compact groups uniform distribution discrepancy theory connected compact abelian groups adeles $p$-adic numbers $p$-adic solenoids dynamical coboundaries cut and project sets $p$-adic internal spaces

Résumé : Bounded remainder sets for a dynamical system are sets for which the Birkhoff averages of return times differ from the expected values by at most a constant amount. These sets are rare and important objects which have been studied for over 100 years. In the last few years there have been a number of results which culminated in explicit constructions of bounded remainder sets for toral rotations in any dimension, of all possible allowable volumes. In this talk we are going to explain these results, and then explain how to generalize them to give explicit constructions of bounded remainder sets for rotations in $p$-adic solenoids. Our method of proof will make use of a natural dynamical encoding of patterns in non-Archimedean cut and project sets.

Codes MSC :
11J71 - Distribution modulo one
11K06 - General theory of distribution modulo 1
11K38 - Irregularities of distribution, discrepancy

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Hennenfent, Guillaume

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https://videos.cirm-math.fr/2017-12-07_Haynes.mp4

Informations sur la Rencontre

Nom de la Rencontre : Jean-Morlet chair: Tiling and recurrence / Chaire Jean-Morlet : Pavages et récurrence
Organisateurs de la Rencontre : Akiyama, Shigeki ; Arnoux, Pierre
Dates : 04/12/2017 - 08/12/2017
Année de la rencontre : 2017
URL de la Rencontre : https://www.chairejeanmorlet.com/1721.html

Données de citation

DOI : 10.24350/CIRM.V.19250803
Citer cette vidéo: Haynes, Alan (2017). Bounded remainder sets for rotations on $p$-adic solenoids. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19250803
URI : http://dx.doi.org/10.24350/CIRM.V.19250803

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Bibliographie

Duneau, M., & Oguey, C. (1990). Displacive transformations and quasicrystalline symmetries. Journal de Physique, 51(1), 5-19 - http://dx.doi.org/10.1051/jphys:019900051010500

Einsiedler, M., & Ward, T. (2011). Ergodic theory with a view towards number theory. London: Springer - http://dx.doi.org/10.1007/978-0-85729-021-2

Grepstad, S., & Lev, N. (2015). Sets of bounded discrepancy for multi-dimensional irrational rotation. Geometric and Functional Analysis, 25(1), 87-133 - http://dx.doi.org/10.1007/s00039-015-0313-z

Haynes, A., Kelly, M., & Koivusalo, H. (2017). Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices. II. Indagationes Mathematicae. New Series, 28(1), 138-144 - http://dx.doi.org/10.1016/j.indag.2016.11.010

Koblitz, N. (1984). $P$-adic numbers, $p$-adic analysis, and zeta-functions. 2nd edition. New York, Heidelberg, Berlin: Springer-Verlag - http://dx.doi.org/10.1007/978-1-4612-1112-9

Weil, A. (1995). Basic number theory. Reprint of the 2nd edition. Berlin: Springer-Verlag - http://dx.doi.org/10.1007/978-3-642-61945-8

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