A Roth type theorem for dense subsets of ${\mathbb{R}}^d$

Abstract

Let $1<p<\infty$, $p\ne 2$. We prove that if $d\ge d_p$ is sufficiently large, and $A\subs\R^d$ is a measurable set of positive upper density then there exists $\la_0=\la_0(A)$ such for all $\la\geq\la_0$ there are $x,y\in {\mathbb{R}}^d$ such that $\{x,x+y,x+2y\}\subs A$ and $|y|_p=\la$, where $||y|{|}_p=({\sum }_i|y_i{|}^p{)}^{1/p}$ is the $l^p({\mathbb{R}}^d)$-norm of a point $y=(y_1,\dots ,y_d)\in {\mathbb{R}}^d$. This means that dense subsets of ${\mathbb{R}}^d$ 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 ${\ell }^{\infty }$-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.

Cited by in corpus (9)

• On side lengths of corners in positive density subsets of the Euclidean space
• Distance Graphs and sets of positive upper density in ${\mathbb{R}}^d$
• A Szemerédi-type theorem for subsets of the unit cube
• Boxes, extended boxes, and sets of positive upper density in the Euclidean space
• Density theorems for anisotropic point configurations
• The density of sets containing large similar copies of finite sets
• Gowers norms control diophantine inequalities
• Coloring and density theorems for configurations of a given volume
• Large dilates of hypercube graphs in the plane