Ultimate periodicity problem for linear numeration systems

Decidability of the HD0L ultimate periodicity problem

In this paper we prove the decidability of the HD0L ultimate periodicity problem.

Numeration Systems, Linear Recurrences, and Regular Sets

A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2, . . . expresses a non-negative integer n as a sum n = ∑i j=0 ajuj . In this case we say the string aiai−1 · · · a1a0 is a representation for n. If gcd(u0, u1, . . .) = g, then every sufficiently large multiple of g has some representation. If the lexicographic ordering on the representations is the ...

Numeration Systems

On multiplicatively dependent linear numeration systems, and periodic points

Two linear numeration systems, with characteristic polynomial equal to the minimal polynomial of two Pisot numbers β and γ respectively, such that β and γ are multiplicatively dependent, are considered. It is shown that the conversion between one system and the other one is computable by a finite automaton. We also define a sequence of integers which is equal to the number of periodic points of...

Conversion between Two Multiplicatively Dependent Linear Numeration Systems