37 research outputs found
Hankel determinants, Pad\'e approximations, and irrationality exponents
The irrationality exponent of an irrational number , which measures the
approximation rate of by rationals, is in general extremely difficult to
compute explicitly, unless we know the continued fraction expansion of .
Results obtained so far are rather fragmentary, and often treated case by case.
In this work, we shall unify all the known results on the subject by showing
that the irrationality exponents of large classes of automatic numbers and
Mahler numbers (which are transcendental) are exactly equal to . Our classes
contain the Thue--Morse--Mahler numbers, the sum of the reciprocals of the
Fermat numbers, the regular paperfolding numbers, which have been previously
considered respectively by Bugeaud, Coons, and Guo, Wu and Wen, but also new
classes such as the Stern numbers and so on. Among other ingredients, our
proofs use results on Hankel determinants obtained recently by Han.Comment: International Mathematics Research Notices 201
Palindromic continued fractions
In the present work, we investigate real numbers whose sequence of partial
quotients enjoys some combinatorial properties involving the notion of
palindrome. We provide three new transendence criteria, that apply to a broad
class of continued fraction expansions, including expansions with unbounded
partial quotients. Their proofs heavily depend on the Schmidt Subspace Theorem
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
Lacunary formal power series and the Stern-Brocot sequence
Let be a real
lacunary formal power series, where and
. It is known that the denominators of
the convergents of its continued fraction expansion are polynomials with
coefficients , and that the number of nonzero terms in is
the th term of the Stern-Brocot sequence. We show that replacing the index
by any 2-adic integer makes sense. We prove that
is a polynomial if and only if . In all the other cases
is an infinite formal power series, the algebraic properties of
which we discuss in the special case .Comment: to appear in Acta Arithmetic