Large subsets of Euclidean space avoiding infinite arithmetic progressions

Abstract: An arithmetic progression (AP) is a collection of equally-spaced real numbers. It may be finite or countably infinite. It is known that if a subset of the real line has positive Lebesgue measure, then it contains a k-term AP for every natural number k. In joint work with Laurestine Bradford (McGill, Linguistics) and Hannah Kohut (UBC, Mathematics), we prove that this result does not extend to infinite APs in the following sense: for each real number p in [0,1), we construct a subset of the real line that intersects every interval of unit length in a set of measure at least p, but that does not contain any infinite AP. In this presentation, I will explain the geometric features of our set that allow it to avoid such progressions. I will also briefly discuss two recent preprints, due to Kolountzakis-Papageorgiou and Burgin-Goldberg-Keleti-MacMahon-Wang, that were inspired by our work. These respectively employ probabilistic and topological methods, in contrast to our argument, which relies on measure theory and equidistribution of sequences mod 1.

analysis of PDEsclassical analysis and ODEsfunctional analysisprobabilityspectral theory

Audience: researchers in the discipline

Quebec Analysis and Related Fields Graduate Seminar