Unique Representations of Real Numbers in Non-Integer Bases

Abstract

Problems related to the expansions of real numbers in non-integer bases have been systematically studied since the late 1950’s, starting with the seminal works by Renyi [16] and Parry [15]. The original approach is based on a specific algorithm for choosing “digits” (e.g. the greedy expansions). This usually leads to the set of sequences of digits for all possible real numbers in question (for instance, non-negative or belonging to a given interval) which, unlike the classical d-adic case, is not a Cartesian product but has a complicated structure. However, in the 1990’s a group of Hungarian mathematicians led by Paul Erdos began to investigate 0-1 sequences that provide unique representations of reals [6, 7, 8]. The present paper continues this line of research. Our set-up is as follows. Let q ∈ (1, 2) be our parameter and Σ = ∏∞1 {0, 1}; we consider those x which have unique expansions in base q of the form