Estimates related to sumfree subsets of sets of integers

A subsetA of the positive integers ?₊ is called sumfree provided (A+A)∩A=ø. It is shown that any finite subsetB of ?₊ contains a sumfree subsetA such that |A|≥1/3(|B|+2), which is a slight improvement of earlier results of P. Erdös Erd] and N. Alon-D. Kleitman A-K]. The method used is harmonic analysis, refining the original approach of Erdös. In general, defines _k (B) as the maximum size of ak-sumfree subsetA ofB, i.e. (A) _k = $\underbrace {A + ... + A}_{k times}$ % MathType!End!2!1! is disjoint fromA. Elaborating the techniques permits one to prove that, for instance, $s_3 \left( B \right) > \frac{{\left| B \right|}}{4} + c\frac{{\log \left| B \right|}}{{\log \log \left| B \right|}}$ % MathType!End!2!1!as an improvement of the estimate $s_k \left( B \right) > \frac{{\left| B \right|}}{4}$ % MathType!End!2!1! resulting from Erdös’ argument. It is also shown that in an inequalitys _k (B)>δ _k |B|, valid for any finite subsetB of ?₊, necessarilyδ _k → 0 fork → ∞ (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.