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 symbolsR,Q,Z, and c stand for the set of real numbers, the set of rational numbers, the set of integers, and the cardinality ofR, respectively. The set of natural numbersf0; 1; 2;:::g will be denoted by either N or !, andjXj will stand for the cardinality of a set X. For A;B R we put A +B =fa +b : a2 A & b2 Bg and LINQ(A) will stand for the linear subspace of R over Q spanned by A. In addition for 0 < n < ! symbol [X] n will stand for the family of all n-element subsets of X