On the distribution of certain Pisot numbers

AbstractA Pisot number θ is said to be simple if the beta-expansion of its fractional part, in base θ, is finite. Let P be the set of such numbers, and S∖P be the complement of P in the set S of Pisot numbers. We show several results about the derived sets of P and of S∖P. A Pisot number θ, with degree greater than 1, is said to be strong, if it has a proper real positive conjugate which is greater than the modulus of the remaining conjugates of θ. The set, say X, of such numbers has been defined by Boyd (1993) [5], and is contained in S∖P. We also prove that the infimum of the j-th derived set of X, where j runs through the set of positive rational integers, is at most j+2