# 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*R*can be determined by considering only subspaces of lower dimension. For example, Stein showed that a set of positive measure in

^{n}*R*, where

^{n}*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

*and supposing that*

**C***λ((A+x)∩*for all reals

**C**)=0*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.