DEPENDENCE OF NUMBERING SYSTEMS ASSOCIATE D WITH THE POWERS OF A PISOT NUMBER

A classical result of Bushi that a set is k-recognizable if and only i f it's k(n)-recognizable. We establish a similar result for number-sys tems associated to Pisot-Vijayaraghavan numbers which satisfy some add itional condition. The theta-expansion of 1 (noted D-theta(1)) is the infinite sequence of positive integers (alpha(n))(n is an element of N ) defined as follows (Bertrand, 1986; Parry, 1960): Let alpha(0) = [th eta], r(0) = {theta}, and for every integer n, alpha(n+1) = [theta r(n )], r(n+1) = {theta r(n)} (where [x] denotes the integral part and {x} the fractional part of x). It's known (Parry, 1960) that P.V.-numbers are ''beta-numbers'' i.e.: either the theta-expansion of 1 is finite (D-0(1) = alpha(0)...alpha(n)), or the theta-expansion of 1 is eventua lly periodic (D-theta(1) = alpha(0) ... alpha(n)(alpha(n+1)...alpha(nm))(omega)). We define a polynomial Q(theta) by Q(theta)(x) = x(n+1) - alpha(0)x(n) - alpha(1)x(n-1) - ... - alpha(n) in the first case, and by Q(theta)(x) = (x(n+m+1) - alpha(0)x(n+m) - alpha(1)x(n+m-1)- ... - alpha(n+m)) - (x(n+1) - (x(n+1) - alpha(0)x(n) - alpha(1)x(n-1) - ... - alpha(n)) in the latter case. theta is a root of Q(theta); when Q(t heta) is the minimal polynomial of theta, we say that theta has the Pi -property (that is always the case for quadratic P.V.-numbers). To the ta is associated a ''theta-number-system'' of the some type as the wel l-known Fibonacci number-system (Knuth, 1968). A subset of integers S is U-theta-recognizable if the language of all finite words given by t he U-theta-coding of integers in S is recognizable by a finite automat on. Our main result: ''Let theta be a P.V.-number. If theta and theta( n)(n greater than or equal to 1) have the Pi-property, then a subset o f integers is U-theta-recognizable if it's U(theta)n-recognizable''.