Résumé

A classical result of Bushi that a set is k-recognizable if and only if it's kⁿ-recognizable. We establish a similar result for number-systems associated to Pisot-Vijayaraghavan numbers which satisfy some additional condition.

The θ-expansion of 1 (noted D_θ(1)) is the infinite sequence of positive integers (α_n)_nεN defined as follows (Bertrand, 1986; Parry, 1960): Let α₀ = [θ], r₀ = {θ}, and for every integer n, α_{n + 1} = [θr_n], r_{n + 1} = {θr_n} (where [x] denotes the integral part and {x} the fractional part of x).

It's known (Parry, 1960) that P.V.-numbers are “β-numbers” i.e.: either the θ-expansion of 1 is finite (D₀(1) = α₀…α_n), or the θ-expansion of 1 is eventually periodic (D_θ(1) = α₀…α_n(α_{n + 1}…α_{n + m})^ω). We define a polynomial Q_θ by Q_θ(x) = x^{n + 1} − α₀xⁿ − α₁x^{n − 1} − … − α_n in the first case, and by Q_θ(x) = (x^{n + m + 1} − α₀x^{n + m} − α₁x^{n + m − 1} − … − α_{n + m}) − (x^{n + 1} − α₀xⁿ − α₁x^{n − 1} − … − α_n) in the latter case. θ is a root of Q_θ; when Q_gq is the minimal polynomial of θ, we say that θ has the Π-property (that is always the case for quadratic P.V.-numbers).

To θ is associated a “θ-number-system” of the some type as the well-known Fibonacci number-system (Knuth, 1968). A subset of integers S is U_θ-recognizable if the language of all finite words given by the U_θ-coding of integers in S is recognizable by a finite automaton.

Our main result: “Let θ be a P.V.-number. If θ and θⁿ(n 1) have the Π-property, then a subset of integers is U_θ-recognizable if it's U_θn-recognizable”.