Large subsets of Euclidean space avoiding infinite arithmetic progressions

TL;DR

The paper constructs large subsets of real numbers that avoid containing any infinite arithmetic progression, showing a contrast to finite progression results in measure theory.

Contribution

It demonstrates the existence of large measure subsets of real numbers that contain no infinite arithmetic progressions, extending understanding of combinatorial structures in measure theory.

Findings

Existence of subsets with positive measure avoiding infinite progressions

Construction method for such subsets for each measure level in [0,1)

Contrasts with finite progression results in measure theory

Abstract

It is known that if a subset of $\mathbb{R}$ has positive Lebesgue measure, then it contains arbitrarily long finite arithmetic progressions. We prove that this result does not extend to infinite arithmetic progressions in the following sense: for each $\lambda$ in $[0,1)$, we construct a subset of $\mathbb{R}$ that intersects every interval of unit length in a set of measure at least $\lambda$, but that does not contain any infinite arithmetic progression.