ESTIMATES RELATED TO SUMFREE SUBSETS OF SETS OF INTEGERS

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. Erdo s [Erd] and N. Alon-D. Kleitman [A-K]. The method used is harmonic ana lysis, 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) = [GR APHICS] 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 argu ment. It is also shown that in an inequality s(k)(B) > delta(k)\B\, va lid for any finite subset B of Z(+), ncessarily delta(k) --> 0 for k - -> infinity (which seemed to be an unclear issue). The most interestin g part of the paper are the methods we believe and the resulting harmo nic analysis questions. They may be worthwhile to pursue.