Strong Measure Zero Sets on the Higher Cantor Space

25.05.2023 11:30 - 13:00

N. Chapman (TU Wien)

As introduced by Borel in the early 20th century, a set of reals is strong measure zero if it can be covered by a sequence of intervals whose lengths shrink arbitrarily fast. This notion admits a natural generalisation in the context of the higher Cantor space \(2^\kappa\). However, contrasting the situation on \(2^\omega\), much about the behaviour of strong measure zero sets on \(2^\kappa\) is unknown; in particular, the consistency of Borel's Conjecture in this context ("A set is strong measure zero iff it has size at most \(\kappa\)") is still open.

We shall discuss a statement closely related to the Borel Conjecture: for \(\kappa\) inaccessible we will sketch the construction of a model of \(|2^\kappa| = \kappa^{++}\) and "\(\forall X \subseteq 2^\kappa: X\) is strong measure zero iff \(|X| \leq \kappa^+\)", focusing on some of the difficulties one runs into when generalising proof strategies from the countable case. Time permitting, we will also briefly touch on Halko's notion of stationary strong measure zero sets.

The content of this talk is extracted from my Master thesis and is based on earlier work by Johannes Schürz.

Please note: There will be a second short talk in this installment of the Set Theory Seminar.




SR 10, 1. Stock, Koling. 14-16, 1090 Wien