  Given a hypersurface $S\subset \mathbb{R}^{2d}$, we study the bilinear
averaging operator that averages a pair of functions over $S$, as well as more
general bilinear multipliers of limited decay and various maximal analogs. Of
particular interest are bilinear maximal operators associated to a fractal
dilation set $E\subset [1,2]$; in this case, the boundedness region of the
maximal operator is associated to the geometry of the hypersurface and various
notions of the dimension of the dilation set. In particular, we determine
Sobolev smoothing estimates at the exponent $L^2 \times L^2 \rightarrow L^2$
using Fourier-analytic methods, which allow us to deduce additional $L^p$
improving bounds for the operators and sparse bounds and their weighted
corollaries for the associated multi-scale maximal functions. We also extend
the method to study analogues of these questions for the triangle averaging
operator and biparameter averaging operators. In addition, some necessary
conditions for boundedness of these operators are obtained.
