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]n will stand for the family of all n-element subsets of X and nA for the n-th algebraic sum of A, that is, nA = n∑ i=1 ai: ai ∈ A for all i = 1, 2,..., n Thus 2S = S + S. For a Polish space X we say that B ⊂ X is a Bernstein set (in X) provided B and X \ B intersect every perfect set P ⊂ X. Clearly every Bernstein subset of an interval in R is neither measurable nor it has the Baire property. For 0 < p < q < ω let Zpq stand for the set of all numbers x ∈ [0, 1] which in base q have representations formed with digits 0,..., p, that is, Zpq = i=1 a(i) qi: a(i) ∈ {0,..., p} for every i = 1, 2,..