We prove an asymptotic for the number of additive triples of bijections {1, . . . , n} → Z/nZ, that is, the number of pairs of bijections π 1 , π 2 : {1, . . . , n} → Z/nZ such that the pointwise sum π 1 + π 2 is also a bijection. This problem is equivalent to counting the number of orthomorphisms or complete mappings of Z/nZ, to counting the number of arrangements of n mutually nonattacking semiqueens on an n × n toroidal chessboard, and to counting the number of transversals in a cyclic Latin square. The method of proof is a version of the Hardy-Littlewood circle method from analytic number theory, adapted to the group (Z/nZ) n .

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Abstract. A well-known conjecture in analytic number theory states that for every pair of sets X, Y Ă Z{pZ, each of size at least log C p (for some constant C) we have that the number of pairs px, yq P XˆY such that x`y is a quadratic residue modulo p differs from 1 2 |X||Y | by o p|X||Y |q. We address the probabilistic analogue of this question, that is for every fixed δ ą 0, given a finite group G and A Ă G a random subset of density 1 2 , we prove that with high probability for all subsets |X|, |Y | ě log 2`δ |G|, the number of pairs px, yq P XˆY such that xy P A differs from

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