We improve the previous work of Yorioka and the first author about the
combinatorics of the ideal SN of strong measure zero sets of reals.
We refine the notions of dominating systems of the first author and introduce
the new combinatorial principle DS(δ) that helps to find simple
conditions to deduce dκ≤cof(SN)
(where dκ is the dominating number on κκ). In
addition, we find a new upper bound of cof(SN) by using
products of relational systems and cardinal characteristics associated with
Yorioka ideals. In addition, we dissect and generalize results from Pawlikowski
to force upper bounds of the covering of SN, particularly for
finite support iterations of precaliber posets.
Finally, as applications of our main theorems, we prove consistency results
about the cardinal characteristics associated with SN and the
principle DS(δ). For example, we show that
cov(SN)<non(SN)=c<cof(SN)
holds in Cohen model, and we refine a result (and the proof) of the first
author about the consistency of
cov(SN)<non(SN)<cof(SN),
with c in any desired position with respect to
cof(SN), and the improvement that
non(SN) can be singular here.Comment: 33 pages, 2 figure