  We consider a problem concerning the distribution of points with missing
digits coordinates that are close to non-degenerate analytic submanifolds. We
show that large enough (to be specified in the paper) sets of points with
missing digits coordinates distribute 'equally' around non-degenerate
submanifolds. As a consequence, we show that intersecting those missing digits
sets with non-degenerate submanifolds always achieve the optimal dimension
reduction. On the other hand, we also prove that there is no lack of points
with missing digits that are contained in non-degenerate submanifolds. Among
the other results,
  1. we prove that the pinned distance sets of those missing digits sets
contain non-trivial intervals regardless of where the pin is.
  2. we prove that for each $\epsilon>0,$ for missing digits sets $K$ with
large bases, simple digit sets (to be specified in the paper), and $\dim_{H}
K>3/4+\epsilon,$ the arithmetic product sets $K\cdot K$ contains non-trivial
intervals.
