Why is the lower bound N/3 and not N/2? Doesn’t the set of all odd numbers make the lower bound N/2?

So cool. On the one hand, going from n/3 to (n/3 + log log n) seems like such a small improvement, but as the article shows, the history is formidable:

- n/3 (Erdős, 1965)

- (n+1)/3 (Alon and Kleitman, 1990)

- (n+2)/3 (Bourgain, 1997)

- n/3 + Ω(log log n) (this paper, Benjamin Bedert, https://arxiv.org/abs/2502.08624)

And the upper bound:

- n/3 + o(n) (Eberhard, Green, Manners, 2014).

Ben Green's list of 100 open problems is which this is (was?) Problem 1, is here: https://people.maths.ox.ac.uk/greenbj/papers/open-problems.p...

Is the main subject here addition or sets?

It's weird to spend paragraphs talking about the incremental improvements to (N+1)/3 and (N+2)/3 and then miss that the new bound is N/3 + c*log(log(n)) for some c>0, not N/3+log(log(n))