We prove that if the Hausdorff dimension of a compact set
is greater
than
,
then the set of three-point configurations determined by
has
positive three-dimensional measure. We establish this by showing that a natural
measure on the set of such configurations has Radon–Nikodym derivative in
if
, and the
index
in this last result cannot, in general, be improved. This problem naturally leads to
the study of a bilinear convolution operator,
where
is surface
measure on the set
,
and we prove a scale of estimates that includes
on
positive functions.
As an application of our main result, it follows that for finite sets of cardinality
and belonging to a natural class of discrete sets in the plane, the
maximum number of times a given three-point configuration arises is
(up to congruence), improving upon the known bound of
in
this context.