Stéphane Fischler

Palindromic prefixes and episturmian words

Let w be an infinite word on an alphabet A. We denote by (ni)i ≥ 1 the increasing sequence (assumed to be infinite) of all lengths of palindromic prefixes of w. In this text, we give an explicit construction of all words w such that ni+1 ≤ 2ni + 1 for all i, and study these words. Special examples include characteristic Sturmian words, and more generally standard episturmian words. As an application, we study the values taken by the quantity lim supni+1/ni, and prove that it is minimal (among all nonperiodic words) for the Fibonacci word.

On the values of G-functions

Let f be a G-function (in the sense of Siegel), and x be an algebraic number; assume that the value f(x) is a real number. As a special case of a more general result, we show that f(x) can be written as g(1), where g is a G-function with rational coefficients and arbitrarily large radius of convergence. As an application, we prove that quotients of such values are exactly the numbers which can be written as limits of sequences a(n)/b(n), where the generating series of both sequences are G-functions with rational coefficients. This result provides a general setting for irrationality proofs in the style of Apery for zeta(3), and gives answers to questions asked by T. Rivoal in [Approximations rationnelles des valeurs de la fonction Gamma aux rationnels : le cas des puissances, Acta Arith. 142 (2010), no. 4, 347-365].

Un exposant de densité en approximation rationnelle

A method and apparatus applicable to jet engines for improving operating efficiency over broad ranges of flight conditions and for reducing engine noise output in take-off and landing by controlling the airflow entering and exiting the engines. A turbojet engine apparatus is described which operates efficiently at both subsonic and supersonic speeds and a method is described which enables a turbofan with an associated satellite turbojet or turbofan to operate more efficiently at both subsonic and supersonic speeds. In both cases, take-off and landing noise is reduced substantially. The apparatus consists essentially of arranging for two separate portions of an engine to act upon one airstream or, alternately, to operate on independent airstreams.

Irrationality exponent and rational approximations with prescribed growth

Let ξ be a real irrational number. We are interested in sequences of linear forms in 1 and ξ, with integer coefficients, which tend to 0. Does such a sequence exist such that the linear forms are small (with given rate of decrease) and the coefficients have some given rate of growth? If these rates are essentially geometric, a necessary condition for such a sequence to exist is that the linear forms are not too small, a condition which can be expressed precisely using the irrationality exponent of ξ. We prove that this condition is actually sufficient, even for arbitrary rates of growth and decrease. We also make some remarks and ask some questions about multivariate generalizations connected to Fischler-Zudilins new proof of Nesterenkos linear independence criterion.

Restricted rational approximation and Apéry-type constructions

Abstract Let ξ be a real irrational number, and φ be a function (satisfying some assumptions). In this text we study the φ-exponenl of irrationality of ξ, defined as the supremum of the set of μ for which there are infinitely many q ≥ 1 such that q is a multiple of φ(q) and | ξ − p q | ≤ q − u for some p ∈ ℤ. We obtain general results on this exponent (a lower bound, the Haussdorff dimension of the set where it is large,…) and connect it to sequences of small linear forms in 1 and ξ with integer coefficients, with geometric behaviour and a divisibility property of the coefficients. Using Aperys proof that ζ(3) is irrational, we obtain an upper bound for the φ-exponent of irrationality of ζ (3), for a given φ.

Interpolation on algebraic groups

We prove an interpolation lemma with multiplicities on a commutative algebraic group. This statement is ‘dual’ to zero estimates used in transcendental number theory. The special case where no multiplicities are involved has been proved by Masser.

Phénomènes de symétrie dans des formes linéaires en polyzêtas

Abstract We give two generalizations, in arbitrary depth, of the symmetry phenomenon used by Ball-Rivoal to prove that infinitely many values of Riemann ζ function at odd integers are irrational. These generalizations concern multiple series of hypergeometric type, which can be written as linear forms in some specific multiple zeta values. The proof makes use of the regularization procedure for multiple zeta values with logarithmic divergence.

The Classical KAM Theorem for Hamiltonian Systems via Rational Approximations

In this paper, we give a new proof of the classical KAM theorem on the persistence of an invariant quasi-periodic torus, whose frequency vector satisfies the Bruno-Rüssmann condition, in real-analytic non-degenerate Hamiltonian systems close to integrable. The proof, which uses rational approximations instead of small divisors estimates, is an adaptation to the Hamiltonian setting of the method we introduced in [4] for perturbations of constant vector fields on the torus.