On the beta expansion of Salem numbers of degree 8

Abstract Boyd showed that the beta expansion of Salem numbers of degree 4 were always eventuallyperiodic. Based on an heuristic argument, Boyd had conjectured that the same is true for Salemnumbers of degree 6 but not for Salem numbers of degree 8. This paper examines Salem numbersof degree 8 and collects experimental evidence in support of Boyd’s conjecture. 1. Introduction and basic de nitionsThe representations of numbers in a non-integer base >1 was pioneered by Renyi [11], wherehe introduced the beta expansion (called also greedy expansion) to represent any real numberxof the interval [0;1] in base by a sequence of digits x 1 x 2 x 3 :::which can be computed bythe following algorithm.Greedy algorithm. Denote by bycand fygthe integer part and the fractional part of areal number y, respectively.Set r 0 = xand for i> 1, x i = b r i 1 c, r i = f r i 1 g.Or, similarly, using the beta transformation T= T of the unit interval which is the mapping:T: [0;1] ! [0;1)x7! xmod(1)where for every i> 1, x