Subsequences of automatic sequences indexed by<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mo stretchy="false">⌊</mml:mo><mml:msup><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">⌋</mml:mo></mml:math>and correlations

Deshouillérs, Jean-Marc; Drmota, Michael; Morgenbesser, Johannes F.

doi:10.1016/j.jnt.2012.03.006

Cited by 13 publications

(20 citation statements)

References 16 publications

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“…There is a quite general theorem for c < 7/5 by Deshoulliers, Drmota, and Morgenbesser [6] (see also [22,23]). …”

Section: Subsequences Along the Sequence N Cmentioning

confidence: 96%

“…The only known result into this direction is the following one due to Deshoulliers, Drmota, and Morgenbesser [6].…”

Section: Theorem 33 ([27]) For Every Non-integermentioning

confidence: 98%

“…We just give here the minimal information for the reader who is not acquainted with those notions (we follow here the presentation of [6]). Definition 2.1.…”

Section: Automatic Sequencesmentioning

confidence: 99%

“…Theorem 3.1 ( [6]). Let q ≥ 2 and u = (u(n)) n≥0 be a q-automatic sequence with values in a finite set E and 1 < c < 7/5.…”

Section: Subsequences Along the Sequence N Cmentioning

confidence: 99%

“…We just mention that by extending the methods of [6] it is possible to prove that for every L ≥ 1 there exists c L > 0 such that every block of length L appears with asymptotic frequency 2 −L in the sequence T ( n c ),…”

Section: Theorem 34 ([6])mentioning

confidence: 99%

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Subsequences of automatic sequences and uniform distribution

Drmota¹

2014

Uniform Distribution and Quasi-Monte Carlo Methods

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Abstract. Automatic sequences and their number theoretic properties have been intensively studied during the last 20 or 30 years. Since automatic sequences are quite regular (they just have linear subword complexity) they cannot be used as quasi-random sequences. However, the situation changes drastically when one uses proper subsequences, for example the subsequence along primes or squares. It is conjectured that the resulting sequences are normal sequences which could be already proved for the Thue-Morse sequence along the subsequence of squares.This kind of research is very challenging and was mainly motivated by the Gelfond problems for the sum-of-digits function. In particular during the last few years there was a spectacular progress due to the Fourier analytic method by Mauduit and Rivat. In this article we survey these recent developments, comment on the proof methods and formulate quite general conjectures. We also present a new result on the subsequence along primes of so-called invertible automatic sequences.

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“…There is a quite general theorem for c < 7/5 by Deshoulliers, Drmota, and Morgenbesser [6] (see also [22,23]). …”

Section: Subsequences Along the Sequence N Cmentioning

confidence: 96%

“…The only known result into this direction is the following one due to Deshoulliers, Drmota, and Morgenbesser [6].…”

Section: Theorem 33 ([27]) For Every Non-integermentioning

confidence: 98%

“…We just give here the minimal information for the reader who is not acquainted with those notions (we follow here the presentation of [6]). Definition 2.1.…”

Section: Automatic Sequencesmentioning

confidence: 99%

“…Theorem 3.1 ( [6]). Let q ≥ 2 and u = (u(n)) n≥0 be a q-automatic sequence with values in a finite set E and 1 < c < 7/5.…”

Section: Subsequences Along the Sequence N Cmentioning

confidence: 99%

Section: Theorem 34 ([6])mentioning

confidence: 99%

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Subsequences of automatic sequences and uniform distribution

Drmota¹

2014

Uniform Distribution and Quasi-Monte Carlo Methods

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Coprimality of Consecutive Elements in a Piatetski-Shapiro Sequence

Deshouillers,

Drmota,

Müllner

2023

Number Theory in Memory of Eduard Wirsing

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Normality of the Thue–Morse sequence along Piatetski-Shapiro sequences, II

Müllner¹,

Spiegelhofer²

2017

Isr. J. Math.

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Abstract. We prove that the Thue-Morse sequence t along subsequences indexed by ⌊n c ⌋ is normal, where 1 < c < 3/2. That is, for c in this range and for each ω ∈ {0, 1} L , where L ≥ 1, the set of occurrences of ω as a factor (contiguous finite subsequence) of the sequence n → t ⌊n c ⌋ has asymptotic density 2 −L . This is an improvement over a recent result by the second author, which handles the case 1 < c < 4/3.In particular, this result shows that for 1 < c < 3/2 the sequence n → t ⌊n c ⌋ attains both of its values with asymptotic density 1/2, which improves on the bound c < 1.4 obtained by Mauduit and Rivat (who obtained this bound in the more general setting of q-multiplicative functions, however) and on the bound c ≤ 1.42 obtained by the second author.In the course of proving the main theorem, we show that 2/3 is an admissible level of distribution for the Thue-Morse sequence, that is, it satisfies a Bombieri-Vinogradov type theorem for each exponent η < 2/3. This improves on a result by Fouvry and Mauduit, who obtained the exponent 0.5924. Moreover, the underlying theorem implies that every finite word ω ∈ {0, 1} L is contained as an arithmetic subsequence of t.

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Subsequences of automatic sequences indexed by⌊nc⌋and correlations

Cited by 13 publications

References 16 publications

Subsequences of automatic sequences and uniform distribution

Subsequences of automatic sequences and uniform distribution

Coprimality of Consecutive Elements in a Piatetski-Shapiro Sequence

Normality of the Thue–Morse sequence along Piatetski-Shapiro sequences, II

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