A strictly ergodic, positive entropy subshift uniformly uncorrelated to the Moebius function

2019/02/11 by Tomasz Downarowicz, Downarowicz, Tomasz, Jacek Serafin +1
Computer Science · Mathematics · #Cellular Automata and Applications #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1902.04162

openalex publication_date 2019/02/11 · openalex created_date 2019/02/21 · openalex updated_date 2026/07/28

Abstract

A recent result of Downarowicz and Serafin (DS) shows that there exist positive entropy subshifts satisfying the assertion of Sarnak's conjecture. More precisely, it is proved that if y=(y_n)_{n≥ 1} is a bounded sequence with zero average along every infinite arithmetic progression (the Möbius function is an example of such a \sq y) then for every N≥ 2 there exists a subshift Σ over N symbols, with entropy arbitrarily close to log N, uncorrelated to y. In the present note, we improve the result of (DS). First of all, we observe that the uncorrelation obtained in (DS) is uniform, i.e., for any continuous function f:Σ→ \mathbb R and every ε>0 there exists n₀ such that for any n≥ n₀ and any x∈Σ we have |\frac1n∑_i=1ⁿf(Tⁱx) y_i|

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