Large subsets avoiding algebraic patterns

Abstract: We prove the existence of a subset of the reals with large sumset and avoiding all linear patterns. This extends a result of K\"orner, who had shown that for any integer $q \geq 1$, there exists a subset $K$ of $\mathbb{R}/\mathbb{Z}$ satisfying no non-trivial linear relations of order $2q-1$ and such that $q.K$ has positive Lebesgue measure. With our method based on transfinite induction, we produce a set with positive measure avoiding all integral linear patterns, at the price of speaking of outer Lebesgue measure. We also discuss questions of measurability of such sets. We then provide constructions of subsets of $\mathbb{R}^n$ avoiding certain non-linear patterns and with maximal Hausdorff dimension.