On two-point configurations in subsets of pseudo- random sets

We prove a transference type result for pseudo-random subsets of
ZN that is analogous to the well-known Furstenberg-S ¨ ark ´ ozy theorem. More pre- ¨
cisely, let k ≥ 2 be an integer and let β and γ be real numbers satisfying
γ + (γ − β)/(2k+1 − 3) > 1.
Let ⊆ ZN be a set with size at least Nγ and linear bias at most Nβ. Then, every
A ⊆ with relative density |A|/|| ≥ (log log N)
− 1
2 log log log log log N contains
a pair of the form {x, x + dk } for some nonzero integer d.
For instance, for squares, i.e., k = 2, and assuming the best possible pseudorandomness β = γ /2 our result applies as soon as γ > 10/11.
Our approach uses techniques of Green as seen in [6] relying on a Fourier
restriction type result also due to Green