On the off chance you read my blog and not Gowers’ — Tim is talking about one of my very favorite open questions, the affine cap set problem, over at his place.

I’m a little ambivalent about reading his posts — every time I think about this problem, I get sucked in and spend a certain amount of time gnawing at it. And the sum total of all this gnawing has so far produced not even the tiniest toothmark on the gleaming surface of the affine cap set problem.

And yet…. and yet…..

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I’ve been tortured by my own seemingly intractable head banger for a long time. It has to with a number of related questions regarding topological properties of some algebraic substructures of the Stone-Cech compactification of infinite discrete groups. Only in this case, a number of years ago I finally actually had a breakthrough and answered one question. It happened one summer evening when I took the bus down toward campus to see a movie. I started jotting some ideas — a possible general approach — in a notebook on the bus, then continued in the movie theater until the lights went out. On the way home I thought it could be workable. It still took about another month or so to fully develop the proofs that I needed, but it all finally fell together. Since then, I’ve concluded that the remaining related questions are probably even more difficult to get traction on. Will the next little breakthrough happen again on the bus? Or while having dinner at my favorite cozy restaurant? Or ever? Sometimes I wonder if the presence of a seemingly natural but intractable problem within a certain framework really means that we are asking that question in the wrong framework. Anyway, I think that mathematics would be a lot less interesting if we didn’t have these seemingly unsurmountable mountains.

Dear Jordan, I have a similar experience with the very same problem. Which direction did you try mainly: improving the lower bounds or the upper bounds?

I’ll try to post about it in the next few days…

Looking forward to it …