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

Brian Cook, Ákos Magyar, Malabika Pramanik

Published 2015-11-18Version 1

Let $A\subset\mathbb{R}^d$ with $d>8$ be a measurable set of positive upper density. We prove that there exists a $\lambda_0=\lambda_0(A)$ such for all $\lambda\geq\lambda_0$ there are $x,y\in\mathbb{R}^d$ such that $\{x,x+y,x+2y\}\subset A$ and $|y|_4=\lambda$, where $|y|_4=(\sum_i y_i^4)^{1/4}$ is the $l^4$-norm of a point $y=(y_1,\ldots,y_d)\in\mathbb{R}^d$. This means that dense subsets of $\mathbb{R}^d$ contain 3-term progressions of arbitrary large gaps when the gap size is measured by the $l^4$-metric. This is known to be false in the ordinary $l^2$-metric and one of the goals of this note is to understand this phenomenon.