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

Brian Cook, Ákos Magyar, Malabika Pramanik

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…