Abstract
Let R R be a commutative ring with identity. In this paper, for a given monotone decreasing positive sequence and an increasing sequence of subsets of R R , we will define a metric on R R using them. Then, we will use this kind of metric to obtain a variant of the proof of Van der Waerden's theorem by Furstenberg [3].
Highlights
Let R be a commutative ring with identity
We will use this kind of metric to obtain a variant of the proof of Van der Waerden’s theorem by Furstenberg [3]
In 1927, Van der Waerden published a famous theorem [5], which states that if the set of positive integers is divided into finitely many classes, at least one of these classes contains arbitrarily long arithmetic progressions
Summary
In 1927, Van der Waerden published a famous theorem [5], which states that if the set of positive integers is divided into finitely many classes, at least one of these classes contains arbitrarily long arithmetic progressions. A dynamical system is defined as a pair (X, T ), where X being a compact metric space and T is a continuous map (homeomorphism) from X into itself. Van der Waerden theorem, dynamical system, metric space.
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