Polynomial configurations in subsets of random and pseudo-random sets

We prove transference results for sparse random and pseudo-random subsets of ZNZN, 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>0C>0 such that for all integer k≥2k≥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+dk}{x,x+dk} for some integer d>0d>0.Let [ZN]p[ZN]p denote the random set obtained by choosing each element from ZNZN with probability p   independently. Our first result shows that for p>N−1/k+o(1)p>N−1/k+o(1) asymptotically almost surely any subset A⊂[ZN]pA⊂[ZN]p (N   prime) of density |A|/pN≥(log⁡N)−15log⁡log⁡log⁡log⁡N contains the polynomial configuration {x,x+dk}{x,x+dk}, 0β>0γ>β>0 be real numbers satisfyingγ+(γ−β)/(2k+1−3)>1.γ+(γ−β)/(2k+1−3)>1. Let Γ⊆ZNΓ⊆ZN (N   prime) be a set of size at least NγNγ and linear bias at most NβNβ. Then our second result implies that every A⊆ΓA⊆Γ with positive relative density contains the polynomial configuration {x,x+dk}{x,x+dk}, 010/11γ>10/11.