The following problem can become a bit of an obsession. I'm curious if there are any serious strategies for attacking it. The problem is a certain Erdos-Turan conjecture.
Let $ B \subseteq {\mathbb N} $. If, for any natural number $k$, we denote by $ r_B(k)$ the number of pairs $(i,j)$ in $B\times B$ such that $i+j=k$.
We call $ B $ an additive basis of order two if $ r_B $ is never $ 0 $.
Erdos-Turan Conjecture for order two bases: If $ B $ is an additive basis of order 2, then $ r_B $ is unbounded.
Are there any serious strategies for attacking this conjecture? If so, what are they?
Application of Szemeredi's theorem quickly handles sets $B$ of positive upper density. The interesting case is the zero upper density case.
The most recent thing I've seen on this is the paper
Sandor, Szaba Csaba A note on a conjecture of Erdos-Turan, INTEGERS: Electronic Journal of Combinatorial Number Theory 8 no. 1 (2008).
(This question may be better for mathstackexchange, but I'm curious if there are any developed lines of attack for research questions like this. Harebrained subquestion: Can Green-Tao type techniques be leveraged for this problem?)
