A Roth‐type theorem for dense subsets of Rd. Issue 4 (29th May 2017) - Cook, Brian
Abstract: Let 1 < p < ∞, p ≠ 2 . We prove that if d ⩾ d p is sufficiently large, and A ⊆ R d is a measurable set of positive upper density then there exists λ 0 = λ 0 ( A ) such that for all λ ⩾ λ 0 there are x, y ∈ R d such that { x, x + y, x + 2 y } ⊆ A and | | y | | p = λ, where | | y | | p = ( ∑ i | y i | p ) 1 / p is the l p ( R d ) ‐norm of a point y = ( y 1, …, y d ) ∈ R d . This means that dense subsets of R d contain 3‐term progressions of all sufficiently large gaps when the gap size is measured in the l p ‐metric. This statement is known to be false in the Euclidean l 2 ‐metric as well as in the l 1 and ℓ ∞ ‐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.