An Approximation Property of Pisot Numbers

✍ Scribed by Vilmos Komornik; Paola Loreti; Marco Pedicini

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 172 KB
• Volume: 80
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1999.2456

✦ Synopsis

Let q>1. Initiated by P. Erdo s et al. in [4], several authors studied the numbers l m (q)=inf [ y: y # 4 m , y{0], m=1, 2, ..., where 4 m denotes the set of all finite sums of the form y== 0 += 1 q+= 2 q 2 + } } } += n q n with integer coefficients &m = i m. It is known ([1], [4], [6]) that q is a Pisot number if and only if l m (q)>0 for all m. The value of l 1 (q) was determined for many particular Pisot numbers, but the general case remains widely open. In this paper we determine the value of l m (q) in other cases.