A note on multiplicative automatic sequences, II

Klurman, Oleksiy; Kurlberg, Pär

doi:10.1112/blms.12318

Cited by 5 publications

(6 citation statements)

References 16 publications

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“…We confirm this conjecture and give some additional structural results. A similar result is also obtained in an upcoming preprint of Klurman and Kurlberg [9].…”

Section: Introductionsupporting

confidence: 83%

“…We confirm this conjecture and give some additional structural results. A similar result is also obtained in an upcoming preprint of Klurman and Kurlberg . Theorem If

a : {double-struckN}_{0} \to C

is an automatic multiplicative sequence, then there exists a threshold

p_{*}

and sequence

χ

which is either a Dirichlet character or identically zero such that

a (n) = χ (n)

for all

n

not divisible any prime

p < p_{*}

.…”

Section: Introductionsupporting

confidence: 60%

See 1 more Smart Citation

On multiplicative automatic sequences

Konieczny

2020

Bull. London Math. Soc.

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“…We confirm this conjecture and give some additional structural results. A similar result is also obtained in an upcoming preprint of Klurman and Kurlberg [9].…”

Section: Introductionsupporting

confidence: 83%

“…We confirm this conjecture and give some additional structural results. A similar result is also obtained in an upcoming preprint of Klurman and Kurlberg . Theorem If

a : {double-struckN}_{0} \to C

is an automatic multiplicative sequence, then there exists a threshold

p_{*}

and sequence

χ

which is either a Dirichlet character or identically zero such that

a (n) = χ (n)

for all

n

not divisible any prime

p < p_{*}

.…”

Section: Introductionsupporting

confidence: 60%

On multiplicative automatic sequences

Konieczny

2020

Bull. London Math. Soc.

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“…Moreover, assume that for p ∈ P large enough the corresponding group extension (T ϕ ) p is also ergodic. 24 Then for p ∈ P large enough, we have…”

Section: Lemma 43 ([13]mentioning

confidence: 99%

“…That said, one realizes immediately that the answer to such a question is negative as periodic functions are automatic sequences and there are many examples of periodic, multiplicative functions. 3 Besides, even amongst non-periodic automatic sequences there are examples of (completely) multiplicative, zero mean functions [2], [40], [43], see also recent [24], [26] for a significant progress on characterizing such sequences. On the other hand, it has been proved in [13] that many automatic sequences given by so-called bijective substitutions are orthogonal to all zero mean, bounded, multiplicative functions (in [9], it is proved that they are orthogonal to the Möbius function).…”

mentioning

confidence: 99%

Automatic sequences are orthogonal to aperiodic multiplicative functions

Lemańczyk

Müllner²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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Given a finite alphabet A and a primitive substitution θ : A → A λ (of constant length λ), let (X θ , S) denote the corresponding dynamical system, where X θ is the closure of the orbit via the left shift S of a fixed point of the natural extension of θ to a self-map of A Z. The main result of the paper is that all continuous observables in X θ are orthogonal to any bounded, aperiodic, multiplicative function u : N → C, i.e. lim N →∞ 1 N n≤N f (S n x)u(n) = 0 for all f ∈ C(X θ) and x ∈ X θ. In particular, each primitive automatic sequence, that is, a sequence read by a primitive finite automaton, is orthogonal to any bounded, aperiodic, multiplicative function.

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“…This has been strengthened in [20] to Weyl almost periodicity (in the primitive case). Relatively recently, multiplicative automatic sequence have been studied in numerous papers, [1,3,4,6,7,12,[17][18][19]21,22,26], and [30]. A particular focus was put on the study of the case of completely multiplicative automatic sequences [1,21] with a complete classification given in [22].…”

Section: Introductionmentioning

confidence: 99%