I recently uploaded “On the largest sum-free subset problem in the integers,” to the arXiv.
Let be a finite subset of the integers. We say is sum-free if there are no solutions to
with . We define to be the size of the largest sum-free subset of . We seek lower bounds for . It is conjectured that
for any . Erdős established is admissible and Bourgain later improved this to . By a construction, Eberhard, Green, and Manners showed that .
I was originally drawn to this problem for two reasons. The first is that the aforementioned result of Erdős is the first additive combinatorics result in Tao and Vu’s additive combinatorics book. The second is that Bourgain’s original proof seemed to have a stigma that it was quite difficult.
We now sketch that is admissible, as shown by Erdős. The first idea is that the set is sum-free. Thus any subset of this set is also sum-free. Note this set has measure , which is the same as the multiplicative constant in .
The second idea is to randomly map into . Indeed choosing
at random, we consider
One can check that this set on average has size and as mentioned before, is sum-free.
Bourgain’s work and also our work involves more careful choices of . Underpinning the work is to think of as a function, , on the torus and to apply a combination of Fourier techniques and combinatorial techniques.
For a set , we let
Then the Erdős argument above may be restated as . Furthermore, would follow from establishing there is an satisfying
One new idea in our work is to partition into and , where is the set of elements in that are divisible by . It turns out that this decomposition is useful as
Thus, for instance, a short argument reveals that if one can establish , then it follows that for .