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

Pinned simplices and connections to product of sets on paraboloids