scholarly journals Sumsets of Finite Beatty Sequences

10.37236/1614 ◽  
2000 ◽  
Vol 8 (2) ◽  
Author(s):  
Jane Pitman
Keyword(s):  
Arithmetic Progression ◽  
Continued Fractions ◽  
Sturmian Word ◽  
Beatty Sequences ◽  
Sturmian Words ◽  
Beatty Sequence

An investigation of the size of $S+S$ for a finite Beatty sequence $S=(s_i)=(\lfloor i\alpha+\gamma \rfloor)$, where $\lfloor \hphantom{x} \rfloor$ denotes "floor", $\alpha$, $\gamma$ are real with $\alpha\ge 1$, and $0\le i \le k-1$ and $k\ge 3$. For $\alpha>2$, it is shown that $|S+S|$ depends on the number of "centres" of the Sturmian word $\Delta S=(s_i-s_{i-1})$, and hence that $3(k-1)\le |S+S|\le 4k-6$ if $S$ is not an arithmetic progression. A formula is obtained for the number of centres of certain finite periodic Sturmian words, and this leads to further information about $|S+S|$ in terms of finite nearest integer continued fractions.

scholarly journals The Maximal Number of Runs in Standard Sturmian Words

10.37236/2473 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Paweł Baturo ◽  
Marcin Piątkowski ◽  
Wojciech Rytter
Keyword(s):  
Explicit Formula ◽  
Special Class ◽  
Linear Time ◽  
Infinite Sequence ◽  
Integer Sequence ◽  
Sturmian Word ◽  
Compact Representations ◽  
Sturmian Words ◽  
Standard Word

We investigate some repetition problems for a very special class $\mathcal{S}$ of strings called the standard Sturmian words, which  have very compact representations in terms of sequences of integers. Usually the size of this word is exponential with respect to the size of its integer sequence, hence we are dealing with repetition problems in compressed strings. An explicit formula is given for the number $\rho(w)$ of runs in a standard word $w$. We show that $\rho(w)/|w|\le 4/5$ for each $w\in S$, and  there is an infinite sequence of strictly growing words $w_k\in {\mathcal{S}}$ such that $\lim_{k\rightarrow \infty} \frac{\rho(w_k)}{|w_k|} = \frac{4}{5}$. Moreover, we show how to compute the number of runs in a standard Sturmian word in linear time with respect to the size of its compressed representation.

scholarly journals Extended Bounds of Beatty Sequence Associated with Primes

2019 ◽  
Vol 8 (6S3) ◽  
pp. 115-118
Keyword(s):  
Character Sums ◽  
Beatty Sequences ◽  
Composite Moduli ◽  
Beatty Sequence

This paper discusses on the estimation of character sums with respect to non-homogeneous Beatty sequences, over prime where , and is irrational. In particular, the bounds is found by extending several properties of character sums associated with composite moduli over prime. As a result, the bound of is deduced.

From Christoffel Words to Markoff Numbers

2018 ◽  
Author(s):  
Christophe Reutenauer
Keyword(s):  
Free Group ◽  
Continued Fractions ◽  
Quadratic Forms ◽  
Special Class ◽  
Rational Numbers ◽  
Real Numbers ◽  
Inner Automorphisms ◽  
Sturmian Words ◽  
Markoff Numbers ◽  
Lyndon Words

Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It characterizes certain quadratic forms, and certain real numbers by extremal inequalities. Both classes are constructed by using certain natural numbers, calledMarkoff numbers; they are characterized by a certain diophantine equality. More basically, they are constructed using certain words, essentially the Christoffel words. The link between Christoffelwords and the theory ofMarkoffwas noted by Frobenius.Motivated by this link, the book presents the classical theory of Markoff in its two aspects, based on the theory of Christoffel words. This is done in Part I of the book. Part II gives the more advanced and recent results of the theory of Christoffel words: palindromes (central words), periods, Lyndon words, Stern–Brocot tree, semi-convergents of rational numbers and finite continued fractions, geometric interpretations, conjugation, factors of Christoffel words, finite Sturmian words, free group on two generators, bases, inner automorphisms, Christoffel bases, Nielsen’s criterion, Sturmian morphisms, and positive automorphisms of this free group.

scholarly journals A Square Root Map on Sturmian Words

10.37236/6074 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Jarkko Peltomäki ◽  
Markus A. Whiteland
Keyword(s):  
Fixed Points ◽  
General Setting ◽  
Square Root ◽  
Sturmian Word ◽  
Word Equation ◽  
Sturmian Words ◽  
Infinite Set ◽  
Curious Property

We introduce a square root map on Sturmian words and study its properties. Given a Sturmian word of slope $\alpha$, there exists exactly six minimal squares in its language (a minimal square does not have a square as a proper prefix). A Sturmian word $s$ of slope $\alpha$ can be written as a product of these six minimal squares: $s = X_1^2 X_2^2 X_3^2 \cdots$. The square root of $s$ is defined to be the word $\sqrt{s} = X_1 X_2 X_3 \cdots$. The main result of this paper is that $\sqrt{s}$ is also a Sturmian word of slope $\alpha$. Further, we characterize the Sturmian fixed points of the square root map, and we describe how to find the intercept of $\sqrt{s}$ and an occurrence of any prefix of $\sqrt{s}$ in $s$. Related to the square root map, we characterize the solutions of the word equation $X_1^2 X_2^2 \cdots X_n^2 = (X_1 X_2 \cdots X_n)^2$ in the language of Sturmian words of slope $\alpha$ where the words $X_i^2$ are minimal squares of slope $\alpha$.We also study the square root map in a more general setting. We explicitly construct an infinite set of non-Sturmian fixed points of the square root map. We show that the subshifts $\Omega$ generated by these words have a curious property: for all $w \in \Omega$ either $\sqrt{w} \in \Omega$ or $\sqrt{w}$ is periodic. In particular, the square root map can map an aperiodic word to a periodic word.

scholarly journals Beatty Sequences, Continued Fractions, and Certain Shift Operators

1976 ◽  
Vol 19 (4) ◽  
pp. 473-482 ◽  
Author(s):  
Kenneth B. Stolarsky
Keyword(s):  
Characteristic Function ◽  
Continued Fractions ◽  
Infinite Sequence ◽  
Positive Root ◽  
Finite Sequence ◽  
Shift Operators ◽  
Beatty Sequences ◽  
Positive Integers

AbstractLet θ = θ(k) be the positive root of θ2 + (k-2)θ-k = 0. Let f(n) = [(n + l)θ]-[nθ] for positive integers n, where [x] denotes the greatest integer in x. Then the elements of the infinite sequence (f(l), f(2), f(3),…) can be rapidly generated from the finite sequence (f(l), f(2),…,f(k)) by means of certain shift operators. For k = 1 we can generate (the characteristic function of) the sequence [nθ] itself in this manner.

scholarly journals Words with many Palindrome Pair Factors

10.37236/5583 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Adam Borchert ◽  
Narad Rampersad
Keyword(s):  
Infinite Words ◽  
Sturmian Word ◽  
Proper Factor ◽  
Sturmian Words

Motivated by a conjecture of Frid, Puzynina, and Zamboni, we investigate infinite words with the property that for infinitely many $n$, every length-$n$ factor is a product of two palindromes. We show that every Sturmian word has this property, but this does not characterize the class of Sturmian words. We also show that the Thue—Morse word does not have this property. We investigate finite words with the maximal number of distinct palindrome pair factors and characterize the binary words that are not palindrome pairs but have the property that every proper factor is a palindrome pair.

scholarly journals Non-residues and primitive roots in Beatty sequences

2006 ◽  
Vol 73 (3) ◽  
pp. 433-443 ◽  
Author(s):  
William D. Banks ◽  
Igor E. Shparlinski
Keyword(s):  
Error Term ◽  
Character Sums ◽  
Multiplicative Character ◽  
Beatty Sequences ◽  
Primitive Roots ◽  
Multiplicative Character Sums ◽  
Beatty Sequence

We study multiplicative character sums taken on the values of a non-homogeneous Beatty sequence where α,β ∈ ℝ, and α is irrational. In particular, our bounds imply that for every fixed ε > 0, if p is sufficiently large and p½+ε ≤ N ≤ p, then among the first N elements of ℬα,β, there are N/2+o(N) quadratic non-residues modulo p. When more information is available about the Diophantine properties of α, then the error term o(N) admits a sharper estimate.

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