Large sets avoiding linear patterns

Abstract

We prove that for any dimension function $h$ with $h\prec x^d$ and for any countable set of linear patterns, there exists a compact set $E$ with ${\mathcal{H}}^h(E)>0$ avoiding all the given patterns. We also give several applications and recover results of Keleti, Maga, and Máthé.

v5: small fixes and clarifications. To appear in Proceedings AMS