Combined algebraic properties of $C^{*}$-sets near zero

Abstract

It is known that for an $IP^*$ set A in ${\mathbb {N}}$ and a sequence $\langle {x_n}\rangle _{n=1}^{\infty }$ there exists a sum subsystem $\langle {y_n}\rangle _{n=1}^{\infty }$ of $\langle {x_n}\rangle _{n=1}^{\infty }$ such that $FS\left( \langle {y_n}\rangle _{n=1}^{\infty }\right)$$\cup$$FP\left( \langle {y_n}\rangle _{n=1}^{\infty }\right) \subseteq A$. Similar types of results also have been proved for central$^*$ and $C^*$-sets where the sequences considered belong to a restricted class of sequences. In 2012, D. De and R. K. Paul have extended these results for $IP^*$ and central$^*$ sets near zero in dense subsemigroups of $((0,\infty ),+)$. In this present work we will extend the results for $C^*$-sets near zero in dense subsemigroups of $((0,\infty ),+)$.