On the successive derived sets of the Pisot numbers

Abstract

Let S denote the set of Pisot (or Pisot-Vijayaraghavan) numbers and let S ( k ) {S^{(k)}} be the kth derived set of S. It is shown that k 1 / 2 ⩽ min S ( k ) {k^{1/2}} \leqslant \min {S^{(k)}} and that lim sup ( min S ( k ) / k ) > 1 \lim \sup (\min {S^{(k)}}/k) > 1 . The lower bound improves the estimate k 1 / 4 ⩽ min S ( k ) {k^{1/4}} \leqslant \min {S^{(k)}} of Dufresnoy and Pisot, while the upper bound improves the obvious estimate min S ( k ) ⩽ k + 1 \min {S^{(k)}} \leqslant k + 1 .