Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals

2015/07/15 by Abdalaoui, El Houcein El, Lemanczyk, Mariusz, De La Rue, Thierry
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1507.04132

Abstract

We show that Sarnak's conjecture on Möbius disjointness holds in every uniquely ergodic modelof a quasi-discrete spectrum automorphism. A consequence of this result is that, for each non constant polynomial P∈\R[x] with irrational leading coefficient and for each multiplicative function \bnu:\N→\C, |\bnu|≤1, we have(1)/(M) ∑_M≤ m\textless2M (1)/(H) | ∑_m≤ n \textless m+H e^2πiP(n)\bnu(n) |\longrightarrow 0 as M→∞, H→∞, H/M→ 0.

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