Betaexpansions

Betaexpansions, or beta expansions, are a way of representing real numbers in a real base β greater than 1. Introduced independently by Alfréd Rényi and W. Parry, this numeration system generalizes ordinary base-b expansions to non-integer bases. For numbers in the unit interval, every x in [0,1] can be written as x = sum_{n=1}^∞ d_n β^{-n}, where the digits d_n range from 0 to ⌊β⌋.

A convenient way to generate a beta-expansion uses the beta transformation Tβ: [0,1) → [0,1), defined by Tβ(x) = βx mod 1. With an initial x0 = x, the digits are given by d_n = ⌊β x_{n-1}⌋ and the orbit x_n = β x_{n-1} − d_n. The greedy expansion uses this rule, producing the lexicographically largest expansion among possible representations. Non-uniqueness can occur for certain x, analogous to the 1 = 0.999... phenomenon in integer bases.

Beta-expansions are closely tied to dynamical systems and symbolic dynamics. The collection of all beta-expansions of numbers in [0,1] forms the beta-shift, a subshift that encodes the combinatorial structure of representations. The special sequence obtained from the expansion of 1, known as the Rényi expansion of 1, governs which digit sequences are admissible.

Notable topics include the study of unique versus multiple expansions, the role of Pisot and other algebraic bases, and extremal questions such as the Komornik–Loreti constant, which concerns bases in which 1 has a unique representation with digits 0 and 1. Betaexpansions have applications in number representation, ergodic theory, and combinatorics on words.