Relation between powers of factors and recurrence function characterizing Sturmian words

In this paper we use the relation of the index of an infinite aperiodic word and its recurrence function to give another characterization of Sturmian words. As a byproduct, we give a new proof of theorem describing the index of a Sturmian word in terms of the continued fraction expansion of its slope. This theorem was independently proved by Carpi and de Luca, and Damanik and Lenz.Comment: 11 page

Self-Matching Properties of Beatty Sequences

We study the selfmatching properties of Beatty sequences, in particular of the graph of the function $\lfloor j\beta\rfloor$ against $j$ for every quadratic unit $\beta\in(0,1)$. We show that translation in the argument by an element $G_i$ of generalized Fibonacci sequence causes almost always the translation of the value of function by $G_{i-1}$. More precisely, for fixed $i\in\N$, we have $\bigl\lfloor \beta(j+G_i)\bigr\rfloor = \lfloor \beta j\rfloor +G_{i-1}$, where $j\notin U_i$. We determine the set $U_i$ of mismatches and show that it has a low frequency, namely $\beta^i$.Comment: 7 page

Integers in number systems with positive and negative quadratic Pisot base

We consider numeration systems with base $\beta$ and $-\beta$, for quadratic Pisot numbers $\beta$ and focus on comparing the combinatorial structure of the sets $\Z_\beta$ and $\Z_{-\beta}$ of numbers with integer expansion in base $\beta$, resp. $-\beta$. Our main result is the comparison of languages of infinite words $u_\beta$ and $u_{-\beta}$ coding the ordering of distances between consecutive $\beta$- and $(-\beta)$-integers. It turns out that for a class of roots $\beta$ of $x^2-mx-m$, the languages coincide, while for other quadratic Pisot numbers the language of $u_\beta$ can be identified only with the language of a morphic image of $u_{-\beta}$. We also study the group structure of $(-\beta)$-integers.Comment: 19 pages, 5 figure

Factor versus palindromic complexity of uniformly recurrent infinite words

We study the relation between the palindromic and factor complexity of infinite words. We show that for uniformly recurrent words one has P(n)+P(n+1) \leq \Delta C(n) + 2, for all n \in N. For a large class of words it is a better estimate of the palindromic complexity in terms of the factor complexity then the one presented by Allouche et al. We provide several examples of infinite words for which our estimate reaches its upper bound. In particular, we derive an explicit prescription for the palindromic complexity of infinite words coding r-interval exchange transformations. If the permutation \pi connected with the transformation is given by \pi(k)=r+1-k for all k, then there is exactly one palindrome of every even length, and exactly r palindromes of every odd length.Comment: 16 pages, submitted to Theoretical Computer Scienc

Number representation using generalized (−β)(-\beta)-transformation

We study non-standard number systems with negative base $-\beta$. Instead of the Ito-Sadahiro definition, based on the transformation $T_{-\beta}$ of the interval $\big[-\frac{\beta}{\beta+1},\frac{1}{\beta+1}\big)$ into itself, we suggest a generalization using an interval $[l,l+1)$ with $l\in(-1,0]$. Such generalization may eliminate certain disadvantages of the Ito-Sadahiro system. We focus on the description of admissible digit strings and their periodicity.Comment: 22 page