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Arbitrarily slow decay in the Möbius disjointness conjecture
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 43 / Issue 9 / September 2023
- Published online by Cambridge University Press:
- 09 September 2022, pp. 2863-2880
- Print publication:
- September 2023
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- Article
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Sarnak’s Möbius disjointness conjecture asserts that for any zero entropy dynamical system
$(X,T)$, 
$({1}/{N})\! \sum _{n=1}^{N}\! f(T^{n} x) \mu (n)= o(1)$ for every 
$f\in \mathcal {C}(X)$ and every 
$x\in X$. We construct examples showing that this 
$o(1)$ can go to zero arbitrarily slowly. In fact, our methods yield a more general result, where in lieu of 
$\mu (n)$, one can put any bounded sequence 
$a_{n}$ such that the Cesàro mean of the corresponding sequence of absolute values does not tend to zero. Moreover, in our construction, the choice of x depends on the sequence 
$a_{n}$ but 
$(X,T)$ does not.
