Polynomial configurations in subsets of random and pseudo-random sets

We prove transference results for sparse random and pseudo-random subsets of ZN, which are analogous to the quantitative version of the well-known Furstenberg-Sárközy theorem due to Balog, Pintz, Steiger and Szemerédi.In the dense case, Balog et al. showed that there is a constant C>0 such that for all integer k≥2 any subset of the first N integers of density at least C(log N)-14log log log log N contains a configuration of the form [x, x+d^k] for some integer d>0.Let [ZN]p denote the random set obtained by choosing each element from ZN with probability p independently. Our first result shows that for p>N^-1/k+o(1) asymptotically almost surely any subset A⊂[ZN]p (N prime) of density |A|/pN≥(log N)-15log log log log N contains the polynomial configuration [x, x+d^k], 0<d≤N^1/k. This improves on a result of Nguyen in the setting of ZN.Moreover, let k≥2 be an integer and let γ>β>0 be real numbers satisfyingγ+(γ-β)/(2k+1-3)>1. Let Γ⊆ZN (N prime) be a set of size at least N^γ and linear bias at most N^β. Then our second result implies that every A⊆Γ with positive relative density contains the polynomial configuration [x, x+d^k], 0<d≤N^1/k.For instance, for squares, i.e., k=. 2, and assuming the best possible pseudo-randomness β. =. γ/2 our result applies as soon as γ > 10/11.