5 research outputs found
Repetitions in infinite palindrome-rich words
Rich words are characterized by containing the maximum possible number of
distinct palindromes. Several characteristic properties of rich words have been
studied; yet the analysis of repetitions in rich words still involves some
interesting open problems. We address lower bounds on the repetition threshold
of infinite rich words over 2 and 3-letter alphabets, and construct a candidate
infinite rich word over the alphabet with a small critical
exponent of . This represents the first progress on an open
problem of Vesti from 2017.Comment: 12 page
The repetition threshold for binary rich words
A word of length is rich if it contains nonempty palindromic factors.
An infinite word is rich if all of its finite factors are rich. Baranwal and
Shallit produced an infinite binary rich word with critical exponent
() and conjectured that this was the least
possible critical exponent for infinite binary rich words (i.e., that the
repetition threshold for binary rich words is ). In this article,
we give a structure theorem for infinite binary rich words that avoid
-powers (i.e., repetitions with exponent at least 2.8). As a consequence,
we deduce that the repetition threshold for binary rich words is
, as conjectured by Baranwal and Shallit. This resolves an open
problem of Vesti for the binary alphabet; the problem remains open for larger
alphabets.Comment: 16 page
On morphisms preserving palindromic richness
It is known that each word of length contains at most distinct
palindromes. A finite rich word is a word with maximal number of palindromic
factors. The definition of palindromic richness can be naturally extended to
infinite words. Sturmian words and Rote complementary symmetric sequences form
two classes of binary rich words, while episturmian words and words coding
symmetric -interval exchange transformations give us other examples on
larger alphabets. In this paper we look for morphisms of the free monoid, which
allow to construct new rich words from already known rich words. We focus on
morphisms in Class . This class contains morphisms injective on the
alphabet and satisfying a particular palindromicity property: for every
morphism in the class there exists a palindrome such that
is a first complete return word to for each letter . We
characterize morphisms which preserve richness over a binary
alphabet. We also study marked morphisms acting on alphabets with
more letters. In particular we show that every Arnoux-Rauzy morphism is
conjugated to a morphism in Class and that it preserves richness
Complementary symmetric Rote sequences: the critical exponent and the recurrence function
We determine the critical exponent and the recurrence function of
complementary symmetric Rote sequences. The formulae are expressed in terms of
the continued fraction expansions associated with the S-adic representations of
the corresponding standard Sturmian sequences. The results are based on a
thorough study of return words to bispecial factors of Sturmian sequences.
Using the formula for the critical exponent, we describe all complementary
symmetric Rote sequences with the critical exponent less than or equal to 3,
and we show that there are uncountably many complementary symmetric Rote
sequences with the critical exponent less than the critical exponent of the
Fibonacci sequence. Our study is motivated by a~conjecture on sequences rich in
palindromes formulated by Baranwal and Shallit. Its recent solution by Curie,
Mol, and Rampersad uses two particular complementary symmetric Rote sequences.Comment: 33 page
Decision Algorithms for Ostrowski-Automatic Sequences
We extend the notion of automatic sequences to a broader class, the Ostrowski-automatic sequences. We develop a procedure for computationally deciding certain combinatorial and enumeration questions about such sequences that can be expressed as predicates in first-order logic.
In Chapter 1, we begin with topics and ideas that are preliminary to this work, including a small introduction to non-standard positional numeration systems and the relationship between words and automata. In Chapter 2, we define the theoretical foundations for recognizing addition in a generalized Ostrowski numeration system and formalize the general theory that develops our decision procedure. Next, in Chapter 3, we show how to implement these ideas in practice, and provide the implementation as an integration to the automatic theorem-proving software package -- Walnut.
Further, we provide some applications of our work in Chapter 4. These applications span several topics in combinatorics on words, including repetitions, pattern-avoidance, critical exponents of special classes of words, properties of Lucas words, and so forth. Finally, we close with open problems on decidability and higher-order numeration systems and discuss future directions for research