Combined additive and multiplicative properties near zero

Abstract

It was proved that whenever ? is partitioned into finitely many cells, one cell must contain arbitrary length geo-arithmetic progressions. It was also proved that arithmetic and geometric progressions can be nicely intertwined in one cell of partition, whenever N is partitioned into finitely many cells. In this article we prove that similar types of results also hold near zero for some suitable dense subsemigroups S of ((0,?),+) for which S?(0,1) is a subsemigroup of ((0,1), middot;).