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.**