On the algebraic sum of a perfect set and a large subset of the reals

In M. Kysiak’s paper "Nonmeasurable algebraic sums of sets of reals", (Coll. Math., Vol. 102, No 1, 2005), the following two questions appeared. Assume that A ⊆ R is a non-meager set with the Baire property and P is perfect. Do there exist meager sets X ⊆ A and Y ⊆ P such that X + Y is not meager? Suppose that A ⊆ R is a measurable set with positive measure and P is perfect. Do there exist measure zero sets X ⊆ A and Y ⊆ P such that X + Y is not null? Identifying R with the Cantor space 2^ω we answer both questions in the positive. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

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