0000000000065073
AUTHOR
Jeffrey Shallit
showing 8 related works from this author
Periodicity, morphisms, and matrices
2003
In 1965, Fine and Wilf proved the following theorem: if (fn)n≥0 and (gn)n≥0 are periodic sequences of real numbers, of period lengths h and k, respectively, and fn = gn for 0 ≤ n > h + k - gcd(h,k), then fn = gn for all n ≥ 0. Furthermore, the constant h + k - gcd(h,k) is best possible. In this paper, we consider some variations on this theorem. In particular, we study the case where fn ≤ gn, instead of fn = gn. We also obtain generalizations to more than two periods.We apply our methods to a previously unsolved conjecture on iterated morphisms, the decreasing length conjecture: if h : Σ* → Σ* is a morphism with |Σ|= n, and w is a word with |w| < |h(w)| < |h2(w)| < ... < |hk(w)|, then k ≤ n.
Variations on a Theorem of Fine & Wilf
2001
In 1965, Fine & Wilf proved the following theorem: if (fn)n≥0 and (gn)n≥0 are periodic sequences of real numbers, of periods h and k respectively, and fn = gn for 0 ≤ n ≤ h+k-gcd(h, k), then fn = gn for all n ≥ 0. Furthermore, the constant h + k - gcd(h, k) is best possible. In this paper we consider some variations on this theorem. In particular, we study the case where fn ≤ gn instead of fn = gn. We also obtain a generalization to more than two periods.
Abelian-Square-Rich Words
2017
An abelian square is the concatenation of two words that are anagrams of one another. A word of length $n$ can contain at most $\Theta(n^2)$ distinct factors, and there exist words of length $n$ containing $\Theta(n^2)$ distinct abelian-square factors, that is, distinct factors that are abelian squares. This motivates us to study infinite words such that the number of distinct abelian-square factors of length $n$ grows quadratically with $n$. More precisely, we say that an infinite word $w$ is {\it abelian-square-rich} if, for every $n$, every factor of $w$ of length $n$ contains, on average, a number of distinct abelian-square factors that is quadratic in $n$; and {\it uniformly abelian-sq…
On lazy representations and Sturmian graphs
2011
In this paper we establish a strong relationship between the set of lazy representations and the set of paths in a Sturmian graph associated with a real number α. We prove that for any non-negative integer i the unique path weighted i in the Sturmian graph associated with α represents the lazy representation of i in the Ostrowski numeration system associated with α. Moreover, we provide several properties of the representations of the natural integers in this numeration system.
On Sturmian Graphs
2007
AbstractIn this paper we define Sturmian graphs and we prove that all of them have a certain “counting” property. We show deep connections between this counting property and two conjectures, by Moser and by Zaremba, on the continued fraction expansion of real numbers. These graphs turn out to be the underlying graphs of compact directed acyclic word graphs of central Sturmian words. In order to prove this result, we give a characterization of the maximal repeats of central Sturmian words. We show also that, in analogy with the case of Sturmian words, these graphs converge to infinite ones.
Sturmian graphs and integer representations over numeration systems
2012
AbstractIn this paper we consider a numeration system, originally due to Ostrowski, based on the continued fraction expansion of a real number α. We prove that this system has deep connections with the Sturmian graph associated with α. We provide several properties of the representations of the natural integers in this system. In particular, we prove that the set of lazy representations of the natural integers in this numeration system is regular if and only if the continued fraction expansion of α is eventually periodic. The main result of the paper is that for any number i the unique path weighted i in the Sturmian graph associated with α represents the lazy representation of i in the Ost…
Sturmian Graphs and a conjecture of Moser
2004
In this paper we define Sturmian graphs and we prove that all of them have a “counting” property. We show deep connections between this counting property and two conjectures, by Moser and by Zaremba, on the continued fraction expansion of real numbers. These graphs turn out to be the underlying graphs of CDAWGs of central Sturmian words. We show also that, analogously to the case of Sturmian words, these graphs converge to infinite ones.
Properties of a Class of Toeplitz Words
2021
We study the properties of the uncountable set of Stewart words. These are Toeplitz words specified by infinite sequences of Toeplitz patterns of the form $\alpha\beta\gamma$, where $\alpha,\beta,\gamma$ is any permutation of the symbols 0,1,?. We determine the critical exponent of the Stewart words, prove that they avoid the pattern $xxyyxx$, find all factors that are palindromes, and determine their subword complexity. An interesting aspect of our work is that we use automata-theoretic methods and a decision procedure for automata to carry out the proofs.