Construction of some Chowla sequences
Shi, Ruxi (2020-10-10)
Shi, R. Construction of some Chowla sequences. Monatsh Math 194, 193–224 (2021). https://doi.org/10.1007/s00605-020-01448-x
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https://urn.fi/URN:NBN:fi-fe2021051930533
Tiivistelmä
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 \((e^{2\pi i \beta ^ng(\beta )})_{n\in {\mathbb {N}}}\) 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 \((e^{2\pi i \beta ^n})_{n\in {\mathbb {N}}}\) 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.
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