taken over all N-element sequences S = (s1, …, sN) of integer elements in a given interval [K + 1, K + L]. In particular, we show that Tm(S, χ) is small on average over all such sequences. We apply it to estimating the number of perfect squares in subsequence sums
in almost all sequences.
We estimate multiplicative character sums over the integers with a fixed sum of binary digits and apply these results to study
the distribution of products of such integers in residues modulo a prime p. Such products have recently appeared in some cryptographic
algorithms, thus our results give some quantitative assurances of their pseudorandomness which is crucial for the security
of these algorithms.