Pablo Shmerkin

Covering the Sierpinski carpet by tubes

A planar set is called tube-null if it can be covered by countably many tubes with the sum of their widths arbitrarily small. By work of Carbery, Soria and Vargas, such sets arise in the localization problem for the Fourier transform. Any set of finite one-dimensional Hausdorff measure is easily seen to be tube-null, but it is often hard to determine whether a given set of dimension larger than 1 is tube-null or not. In particular, there were almost no (non-trivial) examples of tube-null sets of large dimension. I will present our recent result, joint with A. Pyörälä, V. Suomala and M. Wu, that the Sierpinski carpet is tube null. More generally, any times-N invariant set other than the torus is tube-null.