Products of Special Sets of Real Numbers

We develop a 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 meager/null-additive sets in the Cantor space is meager/nulladditive, respectively. (2) 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. (3) The product of a γ-set of reals and a strong measure zero set of reals has strong measure zero. The last assertion solves several problems raised in [14]. 1. Products in the Cantor space The Cantor space C = {0, 1} is equipped with the product topology. For distinct x, y ∈ C, write N(x, y) = min{n : x(n) 6= y(n)}. Then the topology of C is generated by the following metric: