Combined additive and multiplicative properties near zero

摘要

It was proved that whenever N is partitioned into finitelymany cells, one cell must contain arbitrary length geo-arithmeticprogressions. It was also proved that arithmetic and geometricprogressions can be nicely intertwined in one cell of partition,whenever N is partitioned into finitely many cells. Inthis article we prove that similar types of results also hold nearzero for some suitable dense subsemigroups S of((0,∞),+) for which S∩(0,1) is a subsemigroup of((0,1),⋅).