Energy bounds for modular roots and their applications

Abstract: We generalise and improve some recent bounds for additive energies of modular roots. Our arguments use a variety of techniques, including those from additive combinatorics, algebraic number theory and the geometry of numbers. We give applications of these results to new bounds on correlations between {\it Sali{\'e}} sums and to a new equidistribution estimate for the set of modular roots of primes.