Construction of some Chowla sequences
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Construction of some Chowla sequences Ruxi Shi1,2 Received: 22 March 2019 / Accepted: 17 July 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract In this paper, we show that for a twice differentiable function g having countable zeros n and for Lebesgue almost every β > 1, the sequence (e2πiβ g(β) )n∈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 n and show that for Lebesgue almost every β > 1, the sequence (e2πiβ )n∈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. Keywords Chowla conjecture · Sarnak conjecture · Möbius orthogonality · Ergodic theory · Uniform distribution · Stationary process Mathematics Subject Classification Primary: 37A44 · 11J71 · 11N37; Secondary: 37B10
1 Introduction Recall that the Möbius function is an arithmetic function defined by the formula ⎧ ⎪ ⎨1 μ(n) = (−1)k ⎪ ⎩ 0
if n = 1, if n is a product of k distinct primes, otherwise.
Communicated by Adrian Constantin.
B
Ruxi Shi [email protected]
1
Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, 90014 Oulu, Finland
2
´ Present Address: Institute of Mathematics, Polish Academy of Sciences, ul. Sniadeckich 8, 00-656 Warszawa, Poland
123
R. Shi
It is well known that the Möbius function plays an important role in number theory: for example, the statement N
1
μ(n) = O (N 2 + ), for each > 0,
n=1
is equivalent to Riemann’s hypothesis (see [18]). In [2], Chowla formulated a conjecture on the higher orders correlations of the Möbius function: Conjecture 1.1 For each choice of 0 ≤ a1 < · · · < ar , r ≥ 0, i s ∈ {1, 2} not all equal to 2, N −1 1 i0 μ (n + a1 ) · μi1 (n + a2 ) · · · · · μir (n + ar ) = 0. lim N →∞ N n=0
Let T be a continuous transformation on a compact metric space X . Following Sarnak [15], we will say a sequence (a(n))n∈N of complex numbers is orthogonal to the topological dynamical system (X , T ), if lim
N →∞
N −1 1 f (T n x)a(n) = 0, N
(1.1)
n=0
for all x ∈ X and for all f ∈ C(X ) where C(X ) is the space of continuous functions on X . In 2010, Sarnak [15] formulated the following conjecture which is implied by Chowla’s conjecture. Conjecture 1.2 The Möbius function is orthogonal to all the topological dynamical systems of zero entropy. Recently, el Abdalaoui, Kulaga-Przymus, Lema´nczyk and de la Rue [6] studied these conjectures from ergodic theory point of view. Let z be an arbitrary sequence taking values in {−1, 0, 1} as the Möbius function. Following el Abdalaoui et al., we say that z satisfies the condition (Ch) if it satisfies the condition in Chowla’s conjecture1 ; similarly, it satisfies the condition (S0 ) if it satisfies the condition in Sarnak’s conj
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