Abstract
In this paper, we show that for a twice differentiable function g having countable zeros and for Lebesgue almost every beta >1, the sequence (e^{2pi i beta ^ng(beta )})_{nin {mathbb {N}}} is orthogonal to all topological dynamical systems of zero entropy. To this end, we define the Chowla property and the Sarnak property for numerical sequences taking values 0 or complex numbers of modulus 1. We prove that the Chowla property implies the Sarnak property and show that for Lebesgue almost every beta >1, the sequence (e^{2pi i beta ^n})_{nin {mathbb {N}}} shares the Chowla property. It is also discussed whether the samples of a given random sequence have the Chowla property almost surely. Some dependent random sequences having almost surely the Chowla property are constructed.
Highlights
Recall that the Möbius function is an arithmetic function defined by the formula
We show that the Chowla property implies the Sarnak property
We show an example of sequences of independent and identically distributed random variables having the Chowla property almost surely
Summary
Recall that the Möbius function is an arithmetic function defined by the formula. ⎧ ⎪⎨1 μ(n) = ⎪⎩(0−1)k if n = 1, if n is a product of k distinct primes, otherwise. Conjecture 1.2 The Möbius function is orthogonal to all the topological dynamical systems of zero entropy. In [16], we gave several equivalent definitions of oscillating sequences, one of which states that a sequence is fully oscillating if and only if it is orthogonal to all affine maps on all compact abelian groups of zero entropy. We prove that for almost every β > 1, the sequence (e2πiβn )n∈N is a Chowla sequence and is orthogonal to all topological dynamical systems of zero entropy. 5, we will show a unified definition of Chowla sequences and study random sequences: we will give an equivalent condition for stationary random sequences to be Chowla sequences almost surely; we will prove that the Sarnak property does not imply the Chowla property by probability method.
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