Construction of some Chowla sequences

TL;DR

This paper introduces the Chowla property for numerical sequences, proves its implications, and demonstrates that almost all exponential sequences with base greater than one possess this property, linking it to orthogonality in dynamical systems.

Contribution

It establishes the Chowla property for certain sequences, proves it implies the Sarnak property, and constructs dependent random sequences with almost sure Chowla property.

Findings

Almost every exponential sequence with base > 1 has Chowla property.

Chowla property implies Sarnak property.

Dependent random sequences can have almost sure Chowla property.

Abstract

For numerical sequences taking values $0$ or complex numbers of modulus $1$, we define Chowla property and Sarnak property. We prove that Chowla property implies Sarnak property. We also prove that for Lebesgue almost every $\beta>1$, the sequence $(e^{2\pi \beta^n})_{n\in \mathbb{N}}$ shares Chowla property and consequently is orthogonal to all topological dynamical systems of zero entropy. It is also discussed whether the samples of a given random sequence have Chowla property almost surely. Some dependent random sequences having almost surely Chowla property are constructed.