  Topological data analysis (TDA) allows us to explore the topological features
of a dataset. Among topological features, lower dimensional ones have recently
drawn the attention of practitioners in mathematics and statistics due to their
potential to aid the discovery of low dimensional structure in a data set.
However, lower dimensional features are usually challenging to detect based on
finite samples and using TDA methods that ignore the probabilistic mechanism
that generates the data. In this paper, lower dimensional topological features
occurring as zero-density regions of density functions are introduced and
thoroughly investigated. Specifically, we consider sequences of coverings for
the support of a density function in which the coverings are comprised of balls
with shrinking radii. We show that, when these coverings satisfy certain
sufficient conditions as the sample size goes to infinity, we can detect lower
dimensional, zero-density regions with increasingly higher probability while
guarding against false detection. We supplement the theoretical developments
with the discussion of simulated experiments that elucidate the behavior of the
methodology for different choices of the tuning parameters that govern the
construction of the covering sequences and characterize the asymptotic results.
