Measure zero sets whose algebraic sum is non - measurable

In this note we will show that for every natural number n > 0 there exists an S ⊂ [0, 1] such that its n-th algebraic sum nS = S + · · ·+ S is a nowhere dense measure zero set, but its n+1-st algebraic sum nS+S is neither measurable nor it has the Baire property. In addition, the set S will be also a Hamel base, that is, a linear base of R over Q. We use the standard notation as in [2]. Thus symbols R, Q, Z, and c stand for the set of real numbers, the set of rational numbers, the set of integers, and the cardinality of R, respectively. The set of natural numbers {0, 1, 2, . . . } will be denoted by either N or ω, and |X| will stand for the cardinality of a set X. For A,B ⊆ R we put A + B = {a + b : a ∈ A & b ∈ B} and LINQ(A) will stand for the linear subspace of R over Q spanned by A. In addition for 0 < n < ω symbol [X] will stand for the family of all n-element subsets of X and nA for the n-th algebraic sum of A, that is,