2011 seminar talk: Reflecting non-meagreness

Talk held by Juris Steprāns (York University, Canada) at the KGRC seminar on 2011-03-31.

Abstract

A great many results in analysis deal with the question of whether certain properties of sets in Rⁿ can be determined by considering only subspaces of lower dimension. For example, Stein showed that a set of positive measure in Rⁿ, where n≥3 must reflect this property to the surface of a sphere. This was shown later for n=2 by Bourgain and Marstrand. In a similar spirit, Fremlin asked the following: Letting λ be Haar measure on the classical Cantor set C and supposing that λ((A+x)∩C)=0 for all reals x, does it follow that A is Lebesgue null? I will discuss a result obtained with Marton Elekes on the category analogue of this question.