The (−)-integers are natural generalisations of the -integers, and thus of the integers, for negative real bases. When is the analogue of a Parry number, we describe the structure of the set of (−)-integers by a fixed point of an anti-morphism.
The (−)-integers are natural generalisations of the -integers, and thus of the integers, for negative real bases. When is the analogue of a Parry number, we describe the structure of the set of (−)-integers by a fixed point of an anti-morphism.
In the first part of the paper we prove that the Zeckendorf sum-of-digits function and similarly defined functions evaluated on polynomial sequences of positive integers or primes satisfy a central limit theorem. We also prove that the Zeckendorf expansion and the -ary expansions of integers are asymptotically independent.
For abstract numeration systems built on exponential regular languages (including those coming from substitutions), we show that the set of real numbers having an ultimately periodic representation is if the dominating eigenvalue of the automaton accepting the language is a Pisot number. Moreover, if is neither a Pisot nor a Salem number, then there exist points in which do not have any ultimately periodic representation.
Page 1