abstract
An application of van der Waerden's Theorem in additive number theory
Lorenz Halbeisen and Norbert Hungerbühler
A sequence on a finite set of symbols is called strongly
non-repetitive if no two adjacent (finite) segments are
permutations of each other. Replacing the finite set of symbols of a
strongly non-repetitive sequence by different prime numbers, one gets
an infinite sequence on a finite set of integers such that no two
adjacent segments have the same product. It is known that there are
infinite strongly non-repetitive sequences on just four symbols. The
aim of this paper is to show that there is no infinite sequence on a
finite set of integers such that no two adjacent segments have the
same sum. Thus, in the statement above, one cannot replace
''product'' by ''sum''. Further we suggest some strengthened versions
of the notion of strongly non-repetitive.