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