In this note we will show that for every positive natural number n there exists a subset S of [0,1] such that its n-th complex sum nS=S+...+S is a nowhere dense measure zero set, but its (n+1)-th complex 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.
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Last modified March 7, 2001.