The Continuum Hypothesis implies an Erdö-Sierpiński like duality between the ideal of first category subsets of ℝℕ, and the ideal of countable dimensional subsets of ℝℕ. The algebraic sum of a Hurewicz subset - a dimension theoretic analogue of Sierpinski sets and Lusin sets - of ℝℕ with any compactly countable dimensional subset of ℝℕ has first category.
Babinkostova, Liljana and Scheepers, Marion. (2010). "Weakly Infinite Dimensional Subsets of RN". Topology and its Applications, 157(8), 1302-1313. http://dx.doi.org/10.1016/j.topol.2009.03.058