The apparent structure of dense Sidon sets

“What are dense Sidon sets of {1, …, n} like?”, asked Tim Gowers on his blog almost ten years ago. A Sidon set is a set without any solutions to x+y=z+w, which in additive combinatorics jargon means that it has minimal additive structure. Almost paradoxically, large sets with this property appear to be structured in another way, and that’s a bit of a mystery currently.

Now Freddie Manners and I have an idea about the answer to Gowers’s question, at least if “dense” means “really really dense, like 99% as dense as possible”, and the setting is a finite abelian group like \mathbf{Z}/n\mathbf{Z} instead of \{1, \dots, n\}. Our suspicion is that any such set must be related in a specific way to the collineation group of a finite projective plane.

We uploaded our paper to the arxiv today, so have a look and tell us what you think! I also spoke about this recently at CANT 2021. The recording is available here: https://youtu.be/s4ItIkkUvF4

On the other hand, there is also some evidence pointing the other way. Forey and Kowalski showed recently that certain moderately dense Sidon sets arise from algebraic geometry, not from projective planes. It is not clear whether such sets reach the “really really dense” threshold; if so, that would contradict our conjecture, but I suspect they don’t.