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

Abstract

We prove a transference type result for pseudo-random subsets of ℤ_N that is analogous to the well-known Fürstenberg-Sárközy theorem. More precisely, let k ≥ 2 be an integer and let β and γ be real numbers satisfying

$$ \gamma + (\gamma - \beta )/2^{k + 1} - 3) > 1 $$

. Let Γ ⊆ ℤ _N be a set with size at least N ^γ and linear bias at most N ^β. Then, every A ⊆ Γ with relative density $\left| A \right|/\left| \Gamma \right| \geqslant (\log \log N)^{ - \frac{1} {2}\log \log \log \log \log N}$ contains a pair of the form {x, x + d ^k} 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.

Supported by FAPESP (Proc. 2010/16526-3) and by CNPq (Proc. 477203/2012-4) and by NUMEC/USP.