Combined algebraic and multiplicative properties near zero

Details Combined algebraic and multiplicative properties near zero Combined algebraic and multiplicative properties near zero

2012-01-05

arxiv.org/abs/1201.1224v3

Mathematics

Combinatorics

This paper has been withdrawn due to some crutial error

Scientific paper

It was proved that whenever $\mathbb{N}$ is partitioned into finitely many cells, one cell must contain arbitrary length arithmetic and geometric progression nicely intertwined, so that one cell must be rich in the sense of containing substantial combined additive and multiplicative properties. Further it is known that IP$^*$ and central$^*$ sets are also rich in substantial combined additive and multiplicative properties but not partition regular. In this article we prove that these types of results also hold near zero for dense subsemigroups $S$ of $((0,\infty),+)$ for which $(S\cap(0,1),\cdot)$.

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