  In this paper we obtain improved dimensional thresholds for dot product sets
corresponding to compact subsets of a paraboloid. As a direct application of
these estimates, we obtain significant improvements to the best known
dimensional thresholds that guarantee that a given compact subset of Euclidean
space determines a positive proportion of all possible congruence classes of
simplexes. In many regimes this improves the results previously obtained by
Erdogan-Hart-Iosevich (\cite{EHI}), Greenleaf-Iosevich-Liu-Palsson
(\cite{GILP}) and others.
