Additive and multiplicative structure of c$^{\star}$-sets

Abstract

It is known that for an IP${^\star}$ set $A$ in $\mathbb{N}$ and a sequence $< x_{n}>_{n=1}^{\infty}$ there exists a sum subsystem $< y_{n}>_{n=1}^{\infty}$ of $< x_{n}>_{n=1}^{\infty}$ such that $FS(< y_n>_{n=1}^\infty)\cup FP(< y_n>_{n=1}^\infty)\subseteq A$. Similar types of results also have been proved for central* sets where the sequences have been taken from the class of minimal sequences. In this present work we will prove some analogues results for C$^{\star}$-sets for a more general class of sequences.