TY - JOUR

T1 - Polynomial configurations in subsets of random and pseudo-random sets

AU - Aigner-Horev, Elad

AU - Hàn, Hiêp

N1 - Publisher Copyright:
© 2016.

PY - 2016/8/1

Y1 - 2016/8/1

N2 - 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], 01/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], 01/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.

AB - 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], 01/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], 01/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.

KW - Furstenberg-Sárközy Theorem

KW - Polynomial configurations

KW - Pseudo-random subsets

KW - Random subsets

KW - Transference problems

UR - http://www.scopus.com/inward/record.url?scp=84961267083&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2015.12.011

DO - 10.1016/j.jnt.2015.12.011

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AN - SCOPUS:84961267083

SN - 0022-314X

VL - 165

SP - 363

EP - 381

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -