Remarks on gaps in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathrm{Dense}(\mathbb {Q})/\mathbf {nwd}}$\end{document}

The structure \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathrm{Dense}(\mathbb {Q})/\mathbf {nwd}$\end{document} and gaps in analytic quotients of \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$\mathscr {P}(\omega )$\end{document} have been studied in the literature 2, 3, 1. We prove that the structures \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$\mathrm{Dense}(\mathbb {Q})/\mathbf {nwd}$\end{document} and \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$\mathscr {P}(\mathbb {Q})/\mathbf {nwd}$\end{document} have gaps of type \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$(\mathrm{add}(\mathscr {M}), \omega )$\end{document} , and there are no (λ, ω)‐gaps for \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$\lambda < \mathrm{add}(\mathscr {M})$\end{document} , where \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$\mathrm{add}(\mathscr {M})$\end{document} is the additivity number of the meager ideal. We also prove the existence of (ω 1 , ω 1 )‐gaps in these structures. Finally we characterize the cofinality of the meager ideal \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$\mathrm{cof}(\mathscr {M})$\end{document} using families of sets in \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$\mathrm{Dense}(\mathbb {Q})/\mathbf {nwd}$\end{document} .