Description:

Title: Projections and sumsets of self-affine fractals

Abstract: I will discuss some results from our ongoing work with Ian D. Morris which aims at a systematic study of projections of self-affine fractals. After introducing the relevant context, I will discuss
• existence of equilibrium states having non-exact dimensional linear projections (equilibrium states themselves are exact dimensional by Feng);
• existence of self-affine fractals in dimensions at least 4, whose set of exceptional projections in the sense of Marstand Projection Theorem contains higher degree algebraic varieties in Grassmannians (such constructions are not possible even in Borel category in dimension 3 by the solution of a conjecture of Fässler-Orponen by Gan et.al., nor in any dimension if the linear parts of affinities acts strongly irreducibly on all
 exterior powers, by Rapaport);
• existence of self-affine fractals whose sumsets have lower than expected dimension without satisfying an arithmetic resonance (impossible in dimension 1 by Hochman, Shmerkin, Peres and in dimension 2 by Pyorälä).