Confluent linear numeration systems

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

On Properties of Representations in Certain Linear Numeration Systems

Given a ≥ b, let G0 = 1, G1 = a+ 1, and Gn+2 = aGn+1 + bGn for n ≥ 0. For each choice of a and b, we have a linear recurrence that defines a numeration system. Every positive integer n may be written as the sum of the Gn, with alphabet A = {0, 1, . . . a}, in one or more different ways. Let R(a,b)(n) be the function that counts the number of distinct representations of an integer as a sum of th...