Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4593253 | Journal of Number Theory | 2016 | 19 Pages |
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(logN)−14loglogloglogN 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≥(logN)−15loglogloglogN contains the polynomial configuration {x,x+dk}{x,x+dk}, 0