On periodic representations in non-Pisot bases

Abstract

We study periodic expansions in positional number systems with a base β ∈ C, |β| > 1, and with coefficients in a finite set of digits A ⊂ C. We are interested in determining those algebraic bases for which there exists A ⊂ Q(β), such that all elements of Q(β) admit at least one eventually periodic representation with digits in A. In this paper we prove a general result that guarantees the existence of such an A. This result implies the existence of such an A when β is a rational number or an algebraic integer with no conjugates of modulus 1. We also consider periodic representations of elements of Q(β) for which the maximal power of the representation is proportional to the absolute value of the represented number, up to some universal constant. We prove that if every element of Q(β) admits such a representation then β must be a Pisot number or a Salem number. This result generalises a well known result of Schmidt (Bull Lond Math Soc 12(4): [269][270][271][272][273][274][275][276][277][278] 1980).

Periodic representations in algebraic bases

“…It has been shown that the assumption of each element of Q(β) having an eventually periodic (β, A)-representation arising from such a construction is very restrictive. In particular, if β ∈ R, then necessarily |β| is a Pisot number, see [2]. More on finding (β, A)-representations can be found for example in [9,8,1,3,6].…”

“…The following proposition was one of the ingredients used in the proof of Theorem 25 of [2]. Since the proposition appeared as a part of a proof, we include it here for completeness.…”

“…Note that this is a stronger version of Theorem 25 of [2]. An additional condition was required there, namely that 1 a ∈ Z[β, β −1 ], where a is the leading coefficient of the minimal polynomial of β over Z.…”

“…An additional condition was required there, namely that 1 a ∈ Z[β, β −1 ], where a is the leading coefficient of the minimal polynomial of β over Z. This additional condition is not typically satisfied; the full classification of such bases is also given in [2].…”

The Computational Power of Neural Networks and Representations of Numbers in Non-Integer Bases

“…Today's already classical definition of β-expansions was introduced in late 1950s [25,23] and this concept has up to now been a subject of active research, e.g. [1,3,4,5,7,8,9,10,12,18,19,20,24,26,27,28]. The world of β-expansions is much richer than that of the usual integer-base representations as is briefly illustrated on the uniqueness issue of β-expansions.…”