  Several results in the existing literature establish Euclidean density
theorems of the following strong type. These results claim that every set of
positive upper Banach density in the Euclidean space of an appropriate
dimension contains isometric copies of all sufficiently large elements of a
prescribed family of finite point configurations. So far, all results of this
type discussed linear isotropic dilates of a fixed point configuration. In this
paper we initiate the study of analogous density theorems for families of point
configurations generated by anisotropic dilations, i.e., families with
power-type dependence on a single parameter interpreted as their size. More
specifically, here we prove nonisotropic power-type generalizations of a result
by Bourgain on vertices of a simplex, a result by Lyall and Magyar on vertices
of a rectangular box, and a result on distance trees, which is a particular
case of the treatise of distance graphs by Lyall and Magyar. Another source of
motivation for this paper is providing additional evidence for the versatility
of the approach stemming from the work of Cook, Magyar, and Pramanik and its
modification used recently by Durcik and the present author. Finally, yet
another purpose of this paper is to single out anisotropic multilinear singular
integral operators associated with the above combinatorial problems, as they
are interesting on their own.
