Combined algebraic and multiplicative properties near zero

TL;DR

This paper extends the understanding of additive and multiplicative structures in dense subsemigroups near zero, showing that rich combinatorial properties hold in these settings similar to classical results on natural numbers.

Contribution

It proves that combined algebraic and multiplicative properties near zero in dense subsemigroups mirror known results for natural numbers, expanding the scope of such combinatorial phenomena.

Findings

Rich additive and multiplicative structures exist near zero in dense subsemigroups.

Results analogous to classical partition regularity hold near zero.

The properties apply to IP* and central* sets in these semigroups.

Abstract

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 $(S\cap(0,1),\cdot)$.