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Certain results on the Möbius disjointness conjecture
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). (Dynamical Systems)
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

We study certain aspects of the Möbius randomness principle and more specifically the Möbius disjointness conjecture of P. Sarnak. In paper A we establish this conjecture for all orientation preserving circle homeomorphisms and continuous interval maps of zero entropy. In paper B we show, that for all subshifts of finite type with positive topological entropy the Möbius disjointness does not hold. In paper C we study a class of three-interval exchange maps arising from a paper of Bourgain and estimate its Hausdorff dimension. In paper D we consider the Chowla and Sarnak conjectures and the Riemann hypothesis for abstract sequences and study their relationship.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2017. , p. 30
Series
TRITA-MAT-A ; 2017:05
Keywords [en]
Dynamical Systems, Ergodic Theory, Number Theory
National Category
Natural Sciences
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-215682ISBN: 978-91-7729-561-7 (print)OAI: oai:DiVA.org:kth-215682DiVA, id: diva2:1148903
Public defence
2017-11-03, F3, Kungl Tekniska högskolan, Lindstedtsvägen 26,, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

QC 20171016

Available from: 2017-10-16 Created: 2017-10-12 Last updated: 2017-10-16Bibliographically approved
List of papers
1. On Mobius orthogonality for subshifts of finite type with positive topological entropy
Open this publication in new window or tab >>On Mobius orthogonality for subshifts of finite type with positive topological entropy
2017 (English)In: Studia Mathematica, ISSN 0039-3223, E-ISSN 1730-6337, Vol. 237, no 3, p. 277-282Article in journal (Refereed) Published
Abstract [en]

We prove that Mobius orthogonality does not hold for subshifts of finite type with positive topological entropy. This, in particular, shows that all C1+alpha surface diffeomorphisms with positive entropy correlate with the Mobius function.

Place, publisher, year, edition, pages
POLISH ACAD SCIENCES INST MATHEMATICS-IMPAN, 2017
Keywords
subshifts of finite type, entropy, Mobius orthogonality
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-210404 (URN)10.4064/sm8661-10-2016 (DOI)000403153500004 ()2-s2.0-85018502389 (Scopus ID)
Note

QC 20170704

Available from: 2017-07-04 Created: 2017-07-04 Last updated: 2017-10-12Bibliographically approved
2. On Mobius orthogonality for interval maps of zero entropy and orientation-preserving circle homeomorphisms
Open this publication in new window or tab >>On Mobius orthogonality for interval maps of zero entropy and orientation-preserving circle homeomorphisms
2015 (English)In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 53, no 2, p. 317-327Article in journal (Refereed) Published
Abstract [en]

We will prove Sarnak's conjecture on Mobius disjointness for continuous interval maps of zero entropy and also for orientation-preserving circle homeomorphisms by reducing these result to a well-known theorem of Davenport from 1937.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-173755 (URN)10.1007/s11512-014-0208-5 (DOI)000360305500007 ()2-s2.0-84940582441 (Scopus ID)
Note

QC 20150923

Available from: 2015-09-23 Created: 2015-09-18 Last updated: 2017-12-01Bibliographically approved
3. Hausdorff dimension of a class of three-interval exchange maps
Open this publication in new window or tab >>Hausdorff dimension of a class of three-interval exchange maps
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In \cite{B} Bourgain proves that Sarnak's disjointness conjecture holds for a certain class of Three-interval exchange maps. In the present paper we slightly improve the Diophantine condition of Bourgain and estimate the constants in the proof. We further show, that the new parameter set has positive, but not full Hausdorff dimension. This, in particular, implies that the Lebesgue measure of this set is zero.

Keywords
Dynamical Systems, Ergodic Theory, Number Theory
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-215737 (URN)
Note

QC 20171016

Available from: 2017-10-13 Created: 2017-10-13 Last updated: 2017-10-16Bibliographically approved
4. On certain aspects of the Möbius randomness principle
Open this publication in new window or tab >>On certain aspects of the Möbius randomness principle
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper we study different aspects of the Möbius randomness principle. We rephrase the Chowla, Sarnak conjectures and the Riemann hypothesis for abstract sequences and study their relationships. We, in particular, show, that in this setting the Chowla and Sarnak conjectures do not imply the Riemann hypothesis. In the second part of the paper we also study the connection between the multiplicative and additive van der Corput criteria.

Keywords
Number Theory, Dynamical Systems
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-215736 (URN)
Note

QC 20171016

Available from: 2017-10-13 Created: 2017-10-13 Last updated: 2017-10-16Bibliographically approved

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