These distributions form a convex region bounded by hyperplanes, so I just need to prove it for the vertices. For each such vertex, I express it as a linear combination of “triple distributions” (prob distributions on the region with mean (q-1)/3 and all probabilites equal to 0, 1/3, 2/3 or 1) and “reducible distributions” (ones where p_0=p_{q-2}=p_{q-1}=0 and p_1>=p_2>=…>=p_{q-3}).

Since triple distributions are clearly S3-symmetric marginals, and reducible distributions can be converted to things from the q-3 case (by subtracting 1 from everything) we are done by induction.

Devil’s in the details, of course. Hope to write it up this weekend.

]]>The applications to additive number theory are also interesting. I have to assume experts already tried these methods for the cap set problem and got nowhere, but I haven’t tried, so I’ll give it some thought later today.

]]>that’s genuinely fine, keep up writing. ]]>

The agreement says “Prior versions of the article published on non-commercial pre-print servers like arXiv.org can remain on these servers and/or can be updated with Authorâ€™s accepted version.” There is no restriction on the arXiv posting’s license terms. That would lead me to think that CC-BY is acceptable, provided that the article is already posted on arXiv as CC-BY before signing the agreement. ]]>