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

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 \[ |2^κ| = κ^{++} + \forall X \subseteq 2^κ:\ X \text{ is strong measure zero if and only if } |X| \leq κ^+. \] 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.