About the p-paperfolding words

AbstractLet p be an integer greater than or equal to 2. The aim of this paper is to study the language associated to a p-paperfolding sequence. It is known that the number of factors of length n of a 2-paperfolding sequence (i.e. its complexity function) is P(n) = 4n for n ⩾ 7. It is also known that the language of all the factors of all 2-paperfolding sequences is not context-free and that its generating function is transcendental.We show that the complexity function of a p-paperfolding sequence is either strictly subaffine or ultimately linear. The first case never happens if p = 2 or 3. In the second case, the complexity function is either P(n) = 2n or P(n) = 4n for n large enough. We give a simple necessary and sufficient condition for the number of special factors to be p-automatic. We finally show that, for any given p, the language of all factors of all p-paperfolding sequences is not context-free, and that the associated generating series is not algebraic

On tt-extensions of the Hankel determinants of certain automatic sequences

In 1998, Allouche, Peyri\`ere, Wen and Wen considered the Thue--Morse sequence, and proved that all the Hankel determinants of the period-doubling sequence are odd integral numbers. We speak of $t$-extension when the entries along the diagonal in the Hankel determinant are all multiplied by~$t$. Then we prove that the $t$-extension of each Hankel determinant of the period-doubling sequence is a polynomial in $t$, whose leading coefficient is the {\it only one} to be an odd integral number. Our proof makes use of the combinatorial set-up developed by Bugeaud and Han, which appears to be very suitable for this study, as the parameter $t$ counts the number of fixed points of a permutation. Finally, we prove that all the $t$-extensions of the Hankel determinants of the regular paperfolding sequence are polynomials in $t$ of degree less than or equal to $3$

Hankel determinants, Pad\'e approximations, and irrationality exponents

The irrationality exponent of an irrational number $\xi$, which measures the approximation rate of $\xi$ by rationals, is in general extremely difficult to compute explicitly, unless we know the continued fraction expansion of $\xi$. 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 $2$. 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

Continued fractions and transcendental numbers

It is widely believed that the continued fraction expansion of every irrational algebraic number $\alpha$ either is eventually periodic (and we know that this is the case if and only if $\alpha$ is a quadratic irrational), or it contains arbitrarily large partial quotients. Apparently, this question was first considered by Khintchine. A preliminary step towards its resolution consists in providing explicit examples of transcendental continued fractions. The main purpose of the present work is to present new families of transcendental continued fractions with bounded partial quotients. Our results are derived thanks to new combinatorial transcendence criteria recently obtained by Adamczewski and Bugeaud

Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences

We prove that the property of being closed (resp., palindromic, rich, privileged trapezoidal, balanced) is expressible in first-order logic for automatic (and some related) sequences. It therefore follows that the characteristic function of those n for which an automatic sequence x has a closed (resp., palindromic, privileged, rich, trape- zoidal, balanced) factor of length n is automatic. For privileged words this requires a new characterization of the privileged property. We compute the corresponding characteristic functions for various famous sequences, such as the Thue-Morse sequence, the Rudin-Shapiro sequence, the ordinary paperfolding sequence, the period-doubling sequence, and the Fibonacci sequence. Finally, we also show that the function counting the total number of palindromic factors in a prefix of length n of a k-automatic sequence is not k-synchronized