# Yuveshen Mooroogen

# Canada

University of British Columbia

SCS Poster Large Subsets of Euclidean Space Avoiding Infinite Arithmetic Progressions

The question we discuss in our poster is due to Joe Repka (University of Toronto; https://www.math.toronto.edu/cms/people/faculty/repka-joe/). It arose as a

possible extension to a result sometimes attributed to Steinhaus, which

states that every set of positive Lebesgue measure contains arbitrarily

long finite arithmetic progressions. It is clear that sets of infinite

Lebesgue measure need not contain infinite arithmetic progressions:

consider the union of all intervals (n^2, n^2 + 1) for all n in N. In our

poster, we construct an infinite Lebesgue measure set that is a more

"densely packed'' union of intervals (in a sense we make precise) but that

still manages to avoid every infinite arithmetic progression.

https://www.math.ubc.ca/user/3088

http://www.yuveshenmooroogen.com

https://arxiv.org/abs/2205.04786