On synchronized sequences and their separators

Arturo Carpi; Cristiano Maggi

Displaying similar documents to “On synchronized sequences and their separators”

Interpolating sequences in the unit ball of $C^{n}$

N. Pandeski (1985)

Matematički Vesnik

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Lucas sequences with cyclotomic root field

Christian Ballot

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A pair of Lucas sequences Uₙ = (αⁿ-βⁿ)/(α-β) and Vₙ = αⁿ + βⁿ is famously associated with each polynomial x² - Px + Q ∈ ℤ[x] with roots α and β. It is the purpose of this paper to show that when the root field of x² - Px + Q is either ℚ(i), or ℚ(ω), where $ω = e^{2 π i / 6}$ , there are respectively two and four other second-order integral recurring sequences of characteristic polynomial x² - Px + Q that are of the same kinship as the U and V Lucas sequences. These are, when ℚ(α,β) = ℚ(i), the G and the...

A note on univoque self-sturmian numbers

Jean-Paul Allouche (2008)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

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We compare two sets of (infinite) binary sequences whose suffixes satisfy extremal conditions: one occurs when studying iterations of unimodal continuous maps from the unit interval into itself, but it also characterizes univoque real numbers; the other is a disguised version of the set of characteristic sturmian sequences. As a corollary to our study we obtain that a real number $β$ in $(1, 2)$ is univoque and self-sturmian if and only if the $β$ -expansion of $1$ is of the form $1 v$ , where $v$ is a characteristic...

A general upper bound in extremal theory of sequences

Martin Klazar (1992)

Commentationes Mathematicae Universitatis Carolinae

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We investigate the extremal function $f (u, n)$ which, for a given finite sequence $u$ over $k$ symbols, is defined as the maximum length $m$ of a sequence $v = a_{1} a_{2} . . . a_{m}$ of integers such that 1) $1 \leq a_{i} \leq n$ , 2) $a_{i} = a_{j}, i \neq j$ implies $| i - j | \geq k$ and 3) $v$ contains no subsequence of the type $u$ . We prove that $f (u, n)$ is very near to be linear in $n$ for any fixed $u$ of length greater than 4, namely that $f (u, n) = O (n 2^{O (α {(n)}^{| u | - 4})}) .$ Here $| u |$ is the length of $u$ and $α (n)$ is the inverse to the Ackermann function and goes to infinity very slowly. This result extends the estimates in [S] and...

Piatetski-Shapiro sequences via Beatty sequences

Lukas Spiegelhofer (2014)

Acta Arithmetica

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Integer sequences of the form $⌊ n^{c} ⌋$ , where 1 < c < 2, can be locally approximated by sequences of the form ⌊nα+β⌋ in a very good way. Following this approach, we are led to an estimate of the difference $\sum_{n \leq x} φ (⌊ n^{c} ⌋) - 1 / c \sum_{n \leq x^{c}} φ (n) n^{1 / c - 1}$ , which measures the deviation of the mean value of φ on the subsequence $⌊ n^{c} ⌋$ from the expected value, by an expression involving exponential sums. As an application we prove that for 1 < c ≤ 1.42 the subsequence of the Thue-Morse sequence indexed by $⌊ n^{c} ⌋$ attains both of its values with...

On square classes in generalized Fibonacci sequences

Zafer Şiar, Refik Keskin (2016)

Acta Arithmetica

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Let P and Q be nonzero integers. The generalized Fibonacci and Lucas sequences are defined respectively as follows: U₀ = 0, U₁ = 1, V₀ = 2, V₁ = P and $U_{n + 1} = P U ₙ + Q U_{n - 1}$ , $V_{n + 1} = P V ₙ + Q V_{n - 1}$ for n ≥ 1. In this paper, when w ∈ 1,2,3,6, for all odd relatively prime values of P and Q such that P ≥ 1 and P² + 4Q > 0, we determine all n and m satisfying the equation Uₙ = wUₘx². In particular, when k|P and k > 1, we solve the equations Uₙ = kx² and Uₙ = 2kx². As a result, we determine all n such that Uₙ = 6x². ...

Imbalances in Arnoux-Rauzy sequences

Julien Cassaigne, Sébastien Ferenczi, Luca Q. Zamboni (2000)

Annales de l'institut Fourier

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In a 1982 paper Rauzy showed that the subshift $(X, T)$ generated by the morphism $1 \mapsto 12$ , $2 \mapsto 13$ and $3 \mapsto 1$ is a natural coding of a rotation on the two-dimensional torus $𝕋^{2}$ , i.e., is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in $ℝ^{2},$ each domain being translated by the same vector modulo a lattice. It was believed more generally that each sequence of block complexity $2 n + 1$ satisfying a combinatorial criterion known as the $☆$ condition of Arnoux and Rauzy codes the orbit...

On the distance between generalized Fibonacci numbers

Jhon J. Bravo, Carlos A. Gómez, Florian Luca (2015)

Colloquium Mathematicae

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For an integer k ≥ 2, let $(F ₙ^{(k)}) ₙ$ be the k-Fibonacci sequence which starts with 0,..., 0,1 (k terms) and each term afterwards is the sum of the k preceding terms. This paper completes a previous work of Marques (2014) which investigated the spacing between terms of distinct k-Fibonacci sequences.

Discrepancy estimates for some linear generalized monomials

Roswitha Hofer, Olivier Ramaré (2016)

Acta Arithmetica

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We consider sequences modulo one that are generated using a generalized polynomial over the real numbers. Such polynomials may also involve the integer part operation [·] additionally to addition and multiplication. A well studied example is the (nα) sequence defined by the monomial αx. Their most basic sister, ${([n α] β)}_{n \geq 0}$ , is less investigated. So far only the uniform distribution modulo one of these sequences is resolved. Completely new, however, are the discrepancy results proved in this paper....