Summary: REMARKS ON A CONJECTURE ON CERTAIN INTEGER SEQUENCES
SHIGEKI AKIYAMA, HORST BRUNOTTE, ATTILA PETHO, AND WOLFGANG STEINER
Abstract. The periodicity of sequences of integers (an)nZ satisfying the inequalities
0 an-1 + an + an+1 < 1 (n Z)
is studied for real with || < 2. Periodicity is proved in case is the golden ratio; for other
values of statements on possible period lengths are given. Further interesting results on the
morphology of periods are illustrated.
The problem is connected to the investigation of shift radix systems and of Salem numbers.
In this note we will analyze the following conjecture raised in (, Conjecture 6.1):
Conjecture 1.1. Let R and assume that the sequence of integers (an)nZ satisfies the inequal-
(1.1) 0 an-1 + an + an+1 < 1 (n Z).
If || < 2 then (an)nZ is periodic.
The conjecture is supported by extensive computer experiments and by some theorems, which
we will collect below. It is trivially true for = -1, 0, 1.
The conjecture seems to be interesting by itself, but there are also connections to other areas.
Firstly, let us recall the definition of a shift radix system. To a vector r Rd
we associate the
mapping r : Zd