4 research outputs found
An upper bound on asymptotic repetitive threshold of balanced sequences via colouring of the Fibonacci sequence
We colour the Fibonacci sequence by suitable constant gap sequences to
provide an upper bound on the asymptotic repetitive threshold of -ary
balanced sequences. The bound is attained for and and we
conjecture that it happens for infinitely many even 's.
Our bound reveals an essential difference in behavior of the repetitive
threshold and the asymptotic repetitive threshold of balanced sequences. The
repetitive threshold of -ary balanced sequences is known to be at least
for each . In contrast, our bound implies that the
asymptotic repetitive threshold of -ary balanced sequences is at most
for each , where is the golden mean.Comment: arXiv admin note: text overlap with arXiv:2112.0285
The repetition threshold of episturmian sequences
The repetition threshold of a class of infinite -ary sequences is the
smallest real number such that in the class there exists a sequence
that avoids -powers for all . This notion was introduced by Dejean in
1972 for the class of all sequences over a -letter alphabet. Thanks to the
effort of many authors over more than 30 years, the precise value of the
repetition threshold in this class is known for every . The
repetition threshold for the class of Sturmian sequences was determined by
Carpi and de Luca in 2000. Sturmian sequences may be equivalently defined in
various ways, therefore there exist many generalizations to larger alphabets.
Rampersad, Shallit and Vandome in 2020 initiated a study of the repetition
threshold for the class of balanced sequences -- one of the possible
generalizations of Sturmian sequences. Here, we focus on the class of -ary
episturmian sequences -- another generalization of Sturmian sequences
introduced by Droubay, Justin and Pirillo in 2001. We show that the repetition
threshold of this class is reached by the -bonacci sequence and its value
equals , where is the unique positive root of the
polynomial
String attractors of Rote sequences
In this paper, we describe minimal string attractors (of size two) of
pseudopalindromic prefixes of standard complementary-symmetric Rote sequences.
Such a class of Rote sequences forms a subclass of binary generalized
pseudostandard sequences, i.e., of sequences obtained when iterating
palindromic and antipalindromic closures. When iterating only palindromic
closure, palindromic prefixes of standard Sturmian sequences are obtained and
their string attractors are of size two. However, already when iterating only
antipalindromic closure, antipalindromic prefixes of binary pseudostandard
sequences are obtained and we prove that the minimal string attractors are of
size three in this case. We conjecture that the pseudopalindromic prefixes of
any binary generalized pseudostandard sequence have a minimal string attractor
of size at most four