Abstract:We prove among other theorems that it is consistent with $ZFC$ that there exists a set $X\subseteq 2^\omega$ which is not meager additive, yet it satisfies the following property: for each $F_\sigma$ measure zero set $F$, $X+F$ belongs to the intersection ideal $\mathcal M\cap \mathcal N$.
Keywords: $F_\sigma$ measure zero sets; intersection ideal $\mathcal M\cap \mathcal N$; meager additive sets; sets perfectly meager in the transitive sense; $\gamma$-sets
AMS Subject Classification: 03E05 03E17