A Sárközy theorem for finite fields
A classical result of Sárközy states that, for any k∈ ℕ and any positive density subset E of ℕ, there exist elements x and y of E and n ≠ 0 such that x -y = nk. A version of this result for finite fields is derived from a recent theorem of P. Larick, a short proof of which is also given.
Combinatorics Probability and Computing
McCutcheon, R. (2003). A Sárközy theorem for finite fields. Combinatorics Probability and Computing, 12 (5-6 SPEC. ISS.), 643-651. https://doi.org/10.1017/S0963548303005753