More on additive triples of bijections

We study additive properties of the set S of bijections (or permutations) {1, . . . , n} → G, thought of as a subset of Gn, where G is an arbitrary abelian group of order n. Our main result is an asymptotic for the number of solutions to π1 + π2 + π3 = f with π1, π2, π3 ∈ S, where f : {1, . . . , n} → G is an arbitary function satisfying ∑n i=1 f(i) = ∑ G. This extends recent work of Manners, Mrazović, and the author [EMM15]. Using the same method we also prove a less interesting asymptotic for solutions to π1 +π2 +π3 +π4 = f , and we also show that the distribution π1 + π2 is close to flat in L2. As in [EMM15], our method is based on Fourier analysis, and we prove our results by carefully carving up Ĝn and bounding various character sums. This is most complicated when G has even order, say when G = F2. At the end of the paper we explain two applications, one coming from the Latin squares literature (counting transversals in Latin hypercubes) and one from cryptography (PRP-to-PRF conversion).