Harmonic Analysis on the Positive Rationals. Determination of the Group Generated by the Ratios

Abstract

Abstract. The multiplicative group generated by a certain sequence of rationals is determined, settling a 30-year conjecture. §1. Introduction. The present paper settles in the affirmative a 30-yearold conjecture of the first author concerning the representation of an arbitrary positive rational by products and quotients of rationals taken from the sequence (an + b)/(An + B), n = 1, 2, . . . , the fixed coefficient integers a, b, A, B to satisfy a > 0, A > 0, = a B − Ab = 0, cf. [8, Ch. 23, Unsolved Problems 11,12].To appreciate the underlying abstract problem, let Q * denote the multiplicative group of positive rationals, its subgroup generated by a given sequence of positive rationals r n , n = 1, 2, . . . . The extent to which any further positive rational, w, has a representation w = m j=1