A Roth-type theorem for dense subsets of R-d

摘要

Let 1<, p2. We prove that if ddp is sufficiently large, and ARd is a measurable set of positive upper density then there exists 0=0(A) such that for all 0 there are x,yRd such that {x,x+y,x+2y}A and ||y||p=, where ||y||p=(Sigma i|yi|p)1/p is the lp(Rd)-norm of a point y=(y1,...,yd)Rd. This means that dense subsets of Rd contain 3-term progressions of all sufficiently large gaps when the gap size is measured in the lp-metric. This statement is known to be false in the Euclidean l2-metric as well as in the l1 and -metrics. One of the goals of this note is to understand this phenomenon. A distinctive feature of the proof is the use of multilinear singular integral operators, widely studied in classical time-frequency analysis, in the estimation of forms counting configurations.