Strong Measure Zero Sets on $2^\kappa$ for $\kappa$ Inaccessible

TL;DR

This paper explores strong measure zero sets in the higher Cantor space for inaccessible cardinals, establishing consistency results and examining the undecidability of certain measure properties within set theory.

Contribution

It provides two proofs of the relative consistency of measure properties for $2^\kappa$ and investigates the undecidability of stationary strong measure zero in ZFC.

Findings

Consistency of $|2^\kappa|=\kappa^{++}$ with all measure zero sets of size at most $\kappa^+$

Equivalence of strong measure zero and stationary strong measure zero is undecidable in ZFC

Use of perfect tree forcing to analyze measure properties in higher cardinal contexts

Abstract

We investigate the notion of strong measure zero sets in the context of the higher Cantor space $2^\kappa$ for $\kappa$ at least inaccessible. Using an iteration of perfect tree forcings, we give two proofs of the relative consistency of \[ |2^\kappa| = \kappa^{++} + \forall X \subseteq 2^\kappa:\ X \text{ is strong measure zero if and only if } |X| \leq \kappa^+. \] Furthermore, we also investigate the stronger notion of stationary strong measure zero and show that the equivalence of the two notions is undecidable in ZFC.