Substitutions and Möbius disjointness

Ferenczi, Sébastien; Kułaga-Przymus, Joanna; Lemańczyk, Mariusz; Mauduit, Christian

doi:10.1090/conm/678/13645

Cited by 21 publications

(29 citation statements)

References 27 publications

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“…Note that the Sarnak conjecture for a system implies the logarithmic Sarnak conjecture for the same system. The Sarnak conjecture has been proved for a variety of systems, for example nilsystems [34], some horocycle flows [9] and more general zero entropy systems arising from homogeneous dynamics [59], certain distal systems, in particular some extensions of a rotation by a torus [47,51,70], a large class of rank one transformations [3,8,20], systems generated by various substitutions [1,14,19,55], all automatic sequences [57], some interval exchange transformations [8,12,20], some systems of number theoretic origin [7,30], and more... The survey article [18] contains an up to date list of relevant bibliography.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The logarithmic Sarnak conjecture for ergodic weights

2018

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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 which 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.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

The logarithmic Sarnak conjecture for ergodic weights

2018

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“…Their approach was further formalized and generalized in [MR15] (see also [Han17]), but the estimates on the L 1 norm were replaced by a so called "Fourier property" (L ∞ -bounds on the discrete Fourier transform). Finally, the second author recently showed Sarnak's conjecture for automatic sequences [Mül17], generalizing in particular results for synchronizing automatic sequences [DDM15] and invertible automatic sequences [Drm14,FKPLM16].…”

Section: Context and Overviewmentioning

confidence: 92%

Exponential sums with automatic sequences

Drappeau¹,

Müllner²

2018

Acta Arith.

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We show that automatic sequences are asymptotically orthogonal to periodic exponentials of type eq(f (n)), where f is a rational fraction, in the Pólya-Vinogradov range. This applies to Kloosterman sums, and may be used to study solubility of congruence equations over automatic sequences. We obtain this as consequence of a general result, stating that sums over automatic sequences can be bounded effectively in terms of two-point correlation sums over intervals.

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“…See [19], [20] for Sarnak's conjecture. See [1], [2], [6], [9], [14], [17], [15], [23] for some related recent works. The Möbius sequence (µ(n)) is a typical example of fully oscillating sequences ( [5], [13]).…”

Section: Introduction and Resultsmentioning

confidence: 99%