Combinatorial aspects of F$_\sigma$ filters with an application to $\mathcal {N}$-sets

We discuss F$_\sigma$ filters and show that the minimum size of a filter base generating an undiagonalizable filter included in some F$_\sigma$ filter is the better known bounded evasion number ${\frak e}_{ubd}$. An application to $\mathcal {N}$-sets from trigonometric series is given by showing that if $A$ is an $\mathcal {N}$-set and $B$ has size less than ${\frak e}_{ubd}$, then $A \cup B$ is again an ${\cal N}$-set.