Products of special sets of real numbers.

Abstract

We describe a simple machinery which translates results on algebraic sums of sets of reals into the corresponding results on their cartesian product. Some consequences are:

1. The product of a meager/null-additive set and a strong measure zero/strongly meager set in the Cantor space has strong measure zero/is strongly meager, respectively.

2. Using Scheepers' notation for selection principles: $\mathsf{S}_fin(\Omega,\Omega^{gp})\cap\mathsf{S}_1(\mathcal{O},\mathcal{O})=\mathsf{S}_1(\Omega,\Omega^{gp})$, and Borel's Conjecture for $\mathsf{S}_1(\Omega,\Omega)$ (or just $\mathsf{S}_1(\Omega,\Omega^{gp})$) implies Borel's Conjecture.

These results extend results of Scheepers and Miller, respectively.