54 research outputs found
On formal inverse of the Prouhet-Thue-Morse sequence
Let be a prime number and consider a -automatic sequence and its generating function
. Moreover, let us
suppose that and and consider the formal power series
which is a compositional inverse of , i.e.,
. In this note we initiate the study of arithmetic
properties of the sequence of coefficients of the power series . We are
mainly interested in the case when , where
and is the
Prouhet-Thue-Morse sequence defined on the two letter alphabet . More
precisely, we study the sequence which is the
sequence of coefficients of the compositional inverse of the generating
function of the sequence . This sequence is clearly 2-automatic. We
describe the sequence characterizing solutions of the equation
. In particular, we prove that the sequence is 2-regular. We
also prove that an increasing sequence characterizing solutions of the equation
is not -regular for any . Moreover, we present a result
concerning some density properties of a sequence related to .Comment: 16 pages; revised version will appear in Discrete Mathematic
Formal inverses of the generalized Thue-Morse sequences and variations of the Rudin-Shapiro sequence
A formal inverse of a given automatic sequence (the sequence of coefficients
of the composition inverse of its associated formal power series) is also
automatic. The comparison of properties of the original sequence and its formal
inverse is an interesting problem. Such an analysis has been done before for
the Thue{Morse sequence. In this paper, we describe arithmetic properties of
formal inverses of the generalized Thue-Morse sequences and formal inverses of
two modifications of the Rudin{Shapiro sequence. In each case, we give the
recurrence relations and the automaton, then we analyze the lengths of strings
of consecutive identical letters as well as the frequencies of letters. We also
compare the obtained results with the original sequences.Comment: 20 page
Words and Transcendence
Is it possible to distinguish algebraic from transcendental real numbers by
considering the -ary expansion in some base ? In 1950, \'E. Borel
suggested that the answer is no and that for any real irrational algebraic
number and for any base , the -ary expansion of should
satisfy some of the laws that are shared by almost all numbers. There is no
explicitly known example of a triple , where is an integer,
a digit in and a real irrational algebraic number, for
which one can claim that the digit occurs infinitely often in the -ary
expansion of . However, some progress has been made recently, thanks mainly
to clever use of Schmidt's subspace theorem. We review some of these results
Surprises in aperiodic diffraction
Mathematical diffraction theory is concerned with the diffraction image of a
given structure and the corresponding inverse problem of structure
determination. In recent years, the understanding of systems with continuous
and mixed spectra has improved considerably. Moreover, the phenomenon of
homometry shows various unexpected new facets. Here, we report on some of the
recent results in an exemplary and informal fashion.Comment: 9 pages, 1 figure; paper presented at Aperiodic 2009 (Liverpool
Composition inverses of the variations of the Baum-Sweet sequence
Studying and comparing arithmetic properties of a given automatic sequence
and the sequence of coefficients of the composition inverse of the associated
formal power series (the formal inverse of that sequence) is an interesting
problem. This problem was studied before for the Thue-Morse sequence. In this
paper, we study arithmetic properties of the formal inverses of two sequences
closely related to the well-known Baum-Sweet sequence. We give the recurrence
relations for their formal inverses and we determine whether the sequences of
indices at which these formal inverses take value and are regular. We
also show an unexpected connection between one of the obtained sequences and
the formal inverse of the Thue-Morse sequence.Comment: 25 page
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