ε-Pisot numbers in any real algebraic number field are relatively dense

Abstract

An algebraic integer is called an ε-Pisot number ( ε>0) if its Galois conjugates have absolute value less then ε. Let K be any real algebraic number field. We prove that the subset of K consisting of ε-Pisot numbers which have the same degree as that of the field is relatively dense in the real line R . This has some applications to non-stationary products of random matrices involving Salem numbers.