Weakly infinite dimensional subsets of R^N

The Continuum Hypothesis implies an Erd"os-Sierpi'nski like duality between the ideal of first category subsets of ${\mathbb{R}}^{\text{naturals}}$, and the ideal of countable dimensional subsets of ${\mathbb{R}}^{\text{naturals}}$. The algebraic sum of a Hurewicz subset - a dimension theoretic analogue of Sierpinski sets and Lusin sets - of ${\mathbb{R}}^{\text{naturals}}$ with any compactly countable dimensional subset of ${\mathbb{R}}^{\text{naturals}}$ has first category.