On the distribution of certain Pisot numbers

A Pisot number θθ is said to be simple if the beta-expansion of its fractional part, in base θθ, is finite. Let PP be the set of such numbers, and S∖PS∖P be the complement of PP in the set SS of Pisot numbers. We show several results about the derived sets of PP and of S∖PS∖P. A Pisot number θθ, 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 θθ. The set, say XX, of such numbers has been defined by Boyd (1993) [5], and is contained in S∖PS∖P. We also prove that the infimum of the jj-th derived set of XX, where jj runs through the set of positive rational integers, is at most j+2j+2.