# A Freiman-type of question for sets with small doubling costant

I start by saying that I have posted a similar question a few years back, but now I have refined the question a bit more. I have stumbled on this working in finite group theory. The question reminds me a lot Freiman theorem in additive combinatorics.

Let $$n$$ and $$\ell$$ be a positive integers and let $$L$$ be a subset of $$\{1,\ldots,\lfloor (n-1)/2\rfloor\}$$ of cardinality $$\ell$$ with $$\ell\ge cn$$. Here $$c$$ is an absolute constant which does not depend on $$n$$ or $$\ell$$. (I have little information on the actual value of $$c$$.) Let $$B:=\{(x,y,z)\in L\times L\times L\mid x What can be said on $$L$$ if $$|B|> \ell(\ell-1)/4$$? Is the set $$L$$ special, in some arithmetic sense?

Observe that the definition of $$B$$ reminds a lot the definition of additive energy in additive combinatorics, see for instance the book of Tao and Vu.