Liam Orovec, University of Waterloo
Greedy and Lazy expansions for Pisot and Salem Numbers
We call a sequence (a_i) a beta-representation for x provided that the sum over all positive integers ofa_i*beta^{-i} is equal to x. We call the lexicographically largest of of these sequences the greedy expansion for xunder base beta and the smallest we call the lazy expansion. A real number is called a Parry number if its greedyexpansion for 1 is eventually periodic or finite. Similarly we label those real numbers whose lazy expansion for 1is eventually periodic as lazy Parry numbers. Given a PV number with minimal polynomial M(x), we know thatfor sufficiently large value of m that the polynomial T_m^{\pm}(x) has a Salem root. We give criteria that relatesthe greedy and lazy expansions for 1 under these Salem numbers to the greedy and lazy expansions for 1 underthe PV number. We consider Salem numbers of degree 4, the minimal such degree. We will prove these arealways lazy Parry numbers and give explicit constructions for their lazy expansions for 1. We compare theseexpansions to the results of Boyd, that proved that they are also Parry numbers.
MC 5501