Polynomial configurations in subsets of random and pseudo-random sets

Abstract

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+dk} 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+dk}, 0<d≤N1/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 Nγ and linear bias at most Nβ. Then our second result implies that every A⊆Γ with positive relative density contains the polynomial configuration {x,x+dk}, 0<d≤N1/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.