Large subsets avoiding algebraic patterns

TL;DR

The paper develops transfinite-induction methods to construct large subsets of $\mathbb{R}/\mathbb{Z}$ and $\mathbb{R}^n$ that avoid a wide class of algebraic patterns, including all linear relations up to arbitrary order. It proves the existence of an independent set $K$ with full outer measure, but non-measurability, and extends these ideas to Sidon sets with full sumsets and to precise relationships between Hausdorff dimension and sumsets. It also demonstrates the feasibility of high-dimensional pattern-avoidant constructions with full dimension, including algebraic independence, trapezoid-free, angle-free, and distance-free configurations, via a general lemma that unions of few algebraic zero-sets have zero inner measure. Collectively, these results illuminate the tension between measure/dimension and combinatorial pattern avoidance under the axiom of choice, and show that maximal-measure and maximal-dimension pattern-free sets can be achieved in broad settings with nontrivial geometric and algebraic implications.

Abstract

We prove the existence of a subset of the reals with large sumset and avoiding all linear patterns. This extends a result of Körner, 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.

Theorems & Definitions (38)

• Theorem 2.2
• Theorem : Körner, Korner
• Theorem 4.1
• Theorem 5.1
• Theorem 5.2
• Theorem 6.2
• Theorem 6.8
• Definition 1.1
• Definition 1.3
• Lemma 1.4