2003/01/31 by Jean Bourgain, Nets Katz, Terence Tao · 2 citations
Mathematics · #math.CO #math.NT #msc:05B25 #msc:11T99
published as Geom. Func. Anal. 14 (2004), 27-57 · 29 pages. The distance set result needs to be restricted to the case when -1 is not a square
arxiv created 2006/03/10 · arxiv updated 2009/11/30
Let A be a subset of a finite field F := \Z/q\Z for some prime q. If |F|δ< |A| < |F|1-δ for some δ> 0, then we prove the estimate |A+A| + |A.A| ≥ c(δ) |A|1+\eps for some \eps = \eps(δ) > 0. This is a finite field analogue of a result of Erdos and Szemeredi. We then use this estimate to prove a Szemeredi-Trotter type theorem in finite fields, and obtain a new estimate for the Erdos distance problem in finite fields, as well as the three-dimensional Kakeya problem in finite fields.