  Recent work in computability theory has focused on various notions of
asymptotic computability, which capture the idea of a set being "almost
computable." One potentially upsetting result is that all four notions of
asymptotic computability admit "almost computable" sets in every Turing degree
via coding tricks, contradicting the notion that "almost computable" sets
should be computationally close to the computable sets. In response, Astor
introduced the notion of intrinsic density: a set has defined intrinsic density
if its image under any computable permutation has the same asymptotic density.
Furthermore, Astor also introduced various notions of intrinsic computation in
which the standard coding tricks cannot be used to embed intrinsically
computable sets in every Turing degree. Our goal is to study the sets which are
intrinsically small, i.e. those that have intrinsic density zero. We begin by
studying which computable functions preserve intrinsic smallness. We also show
that intrinsic smallness and hyperimmunity are computationally independent
notions of smallness, i.e. any hyperimmune degree contains a Turing-equivalent
hyperimmune set which is "as large as possible" and therefore not intrinsically
small. Our discussion concludes by relativizing the notion of intrinsic
smallness and discussing intrinsic computability as it relates to our study of
intrinsic smallness.
