Preprint C132/2011
Dynamics of sets of zero density
Yuri Lima
Keywords: zero density | Marstrand's theorem | Z^d-actions
This thesis is devoted to the study of sets of zero density, both continuous - inside R - and discrete - inside Z. Most of the arguments are combinatorial in nature. The interests are twofold. The first investigates arithmetic sums: given two subsets E,F of R or Z, what can be said about E+cF for most of the parameters c in R? The continuous case dates back to Marstrand. We give an alternative combinatorial proof of his theorem for the particular case of products of regular Cantor sets and, in the sequel, extend these techniques to give the proof of the general case. We also discuss Marstrand's theorem in discrete spaces. More specifically, we propose a fractal dimension for subsets of the integers and establish a Marstrand type theorem in this context. The second interest is ergodic theoretical. It contains constructions of Z^d-actions with prescribed topological and ergodic properties, such as total minimality, total ergodicity and total strict ergodicity. These examples prove that Bourgain's polynomial pointwise ergodic theorem has not a topological version.