Additive Patterns in Multiplicative Subgroups

The study of sum and product problems in finite fields motivates the investigation of additive structures in multiplicative subgroups of such fields. A simple known fact is that any multiplicative subgroup of size at least q3/4 in the finite field Fq must contain an additive relation x + y = z. Our main result is that there are infinitely many examples of sum-free multiplicative subgroups of size Ω(p1/3) in prime fields Fp. More complicated additive relations are studied as well. One representative result is the fact that the elements of any multiplicative subgroup H of size at least q3/4+o(1) of Fq can be arranged in a cyclic permutation so that the sum of any pair of consecutive elements in the permutation belongs to H. The proofs combine combinatorial techniques based on the spectral properties of Cayley sum-graphs with tools from algebraic and analytic number theory.