# Large sets avoiding linear patterns

## TL;DR

This paper constructs large fractal sets that avoid specified linear patterns, extending previous results and providing new applications in geometric measure theory.

## Contribution

It introduces a method to build compact sets with positive measure avoiding countably many linear patterns, generalizing earlier findings.

## Key findings

- Existence of large sets avoiding countably many linear patterns
- Construction of sets with positive measure under certain dimension functions
- Recovery of previous results by Keleti, Maga, and Máthé

## 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 M\'{a}th\'{e}.