A subset A of the positive integers Z(+) is called sumfree provided (A + A) boolean AND A = 0. It is shown that any finite subset B of Z(+) contains a sumfree subset A such that \A\ greater than or equal to 1/3(\B\ + 2), which is a slight improvement of earlier results of P. Erdos [Erd] and N. Alon-D. Kleitman [A-K]. The method used is harmonic analysis, refining the original approach of Erdos. In general, define sk(B) as the maximum size of a k-sumfree subset A of B, i.e. (A)(k) = [GRAPHICS] is disjoint from A. Elaborating the techniques permits one to prove that, for instance, s(3)(B) > \B\/4 + c log\B\/log log \B\ as an improvement of the estimate s(k)(B) > \B\/4 resulting from Erdos argument. It is also shown that in an inequality s(k)(B) > delta(k)\B\, valid for any finite subset B of Z(+), ncessarily delta(k) --> 0 for k --> infinity (which seemed to be an unclear issue). The most interesting part of the paper are the methods we believe and the resulting harmonic analysis questions. They may be worthwhile to pursue.
