Additive Patterns in Multiplicative Subgroups

Abstract

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 q 3/4 in the finite field F q 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(p 1/3 ) in prime fields F p . 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 q 3/4+o(1) of F q 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.

Powers of Hamilton Cycles in Pseudorandom Graphs

“…The proof of this theorem amounts to showing that a certain graph on vertex set A is pseudorandom and applying the Hamiltonicity result of Krivelevich and Sudakov [23] to find a Hamilton cycle in this graph, which defines the cyclic ordering. We can replace that result with Corollary 4 to obtain the following result, which in particular strengthens Proposition 1.6 of [4].…”

“…Theorem 23 (Alon and Bourgain [4], Theorem 1.2). There exists an absolute positive constant c such that for any prime power q and for any multiplicative subgroup A of the finite field F q of size |A| = d ≥ c q 3/4 (log q) 1/2 (log log log q) 1/2 log log q there is a cyclic ordering a 0 , a 1 , .…”

A note on the set A(A + A)

“…This is related to the question of how large a sum-free multiplicative subgroup of F * p can be. Alon and Bourgain showed [2] that it can be at least Ω(p 1/3 ). For fixed s, t ∈ S, we have…”

Arithmetic progressions in multiplicative groups of finite fields

“…This last result is simple and well-known, but though one could conjecture a condition of the form |G| > cp 1 2 to suffice, still the best available in this direction. (See [1] for instance. )…”