A variant of the proof of Van der Waerden's theorem by Furstenberg

Abstract

Let $R$ be a commutative ring with identity. In this paper, for a given monotone decreasing positive sequence and an increasing sequence of subsets of $R$, we will define a metric on $R$ 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].

Keywords

• Van der Waerden Theorem
• dynamical system
• metric space

References

1. Birkhoff, G.D., Dynamical Systems, Math. Soc. Coll. Publ., vol 9, Amer. Math. Soc., Providence RI, 1927. https://doi.org/http://dx.doi.org/10.1090/coll/009
2. Engelking, R., General Topology, Second Edition, Heldermann Verlag, Berlin, 1989. Furstenberg, H., Poincare recurrence and number theory, Bull. of the A. Math. Soc., 5 (3) (1981), 211–234.
3. Furstenberg, H., Recurrence in Ergodic Theory and Combinatorial Number, Princeton University Press, Princeton, New Jersey, 1981.
4. Van der Waerden, B.L., Beweis einer baudetschen vermutung, Nieuw Arch. Wisk., 15 (1927), 212–216.