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The Möbius disjointness conjecture of Sarnak states that the Möbius function does not correlate with any bounded sequence of complex numbers arising from a topological dynamical system with zero topological entropy. We verify the logarithmically averaged variant of this conjecture for a large class of systems, which includes all uniquely ergodic systems with zero entropy. One consequence of our results is that the Liouville function has super-linear block growth. Our proof uses a disjointness argument, and the key ingredient is a structural result for measure preserving systems naturally associated with the Möbius and the Liouville function. We prove that such systems have no irrational spectrum and their building blocks are infinite-step nilsystems and Bernoulli systems. To establish this structural result we make a connection with a problem of purely ergodic nature via some identities recently obtained by Tao. In addition to an ergodic structural result of Host and Kra, our analysis is guided by the notion of strong stationarity that was introduced by Furstenberg and Katznelson in the early 90’s and naturally plays a central role in the structural analysis of measure preserving systems associated with multiplicative functions.
Nikos Frantzikinakis 1 ; Bernard Host 2
@article{10_4007_annals_2018_187_3_6,
author = {Nikos Frantzikinakis and Bernard Host},
title = {The logarithmic {Sarnak} conjecture for ergodic weights},
journal = {Annals of mathematics},
pages = {869--931},
publisher = {mathdoc},
volume = {187},
number = {3},
year = {2018},
doi = {10.4007/annals.2018.187.3.6},
mrnumber = {3779960},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2018.187.3.6/}
}
TY - JOUR AU - Nikos Frantzikinakis AU - Bernard Host TI - The logarithmic Sarnak conjecture for ergodic weights JO - Annals of mathematics PY - 2018 SP - 869 EP - 931 VL - 187 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2018.187.3.6/ DO - 10.4007/annals.2018.187.3.6 LA - en ID - 10_4007_annals_2018_187_3_6 ER -
%0 Journal Article %A Nikos Frantzikinakis %A Bernard Host %T The logarithmic Sarnak conjecture for ergodic weights %J Annals of mathematics %D 2018 %P 869-931 %V 187 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2018.187.3.6/ %R 10.4007/annals.2018.187.3.6 %G en %F 10_4007_annals_2018_187_3_6
Nikos Frantzikinakis; Bernard Host. The logarithmic Sarnak conjecture for ergodic weights. Annals of mathematics, Tome 187 (2018) no. 3, pp. 869-931. doi: 10.4007/annals.2018.187.3.6
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