Comments on: EDP19 — removing some vagueness

By: Alec Edgington

Alec Edgington — Fri, 10 Sep 2010 20:01:50 +0000

Gil also mentioned this a while back. It does look rather similar -- though the impression I get from a skim through that article of Casazza and Edidin is that the problem has more of an algebraic and less of a number-theoretic flavour. (For example, the word 'prime' doesn't occur anywhere in that article, whereas it's hard to imagine a discussion of EDP that didn't talk about primes at some point.) But it could well be that we could learn something from approaches to the Kadison--Singer problem.

By: gowers

gowers — Thu, 09 Sep 2010 18:09:46 +0000

Indeed, if you look along the even numbers, you find that the first one (and possibly the second — I haven’t checked) has discrepancy a lot more than 3. This makes me think you may have some misconception about the definition of discrepancy.

By: Klas Markström

Klas Markström — Thu, 09 Sep 2010 18:01:44 +0000

Jason, your sequences start with three 1:s, so they have discrepancy at least 3.

By: Jason

Jason — Thu, 09 Sep 2010 00:04:30 +0000

Not directly related to this post, but I am curious if experimental results of long sequences of discrepancy 2 are still interesting/relevant/helpful. I thought about it over the weekend after reading the EDP18 post and checking out the polymath wiki, and managed to come up with a method to generate such sequences, assuming I am computing the discrepancy correctly, and not overlooking some desired structure that makes the problem harder. Found a sequence of length 2026 that is relatively uniform here: http://pastebin.com/S3KZVx4V Another sequence of length 20386 that isn’t very uniform was found here: http://pastebin.com/fUwr6tg0

By: DC

DC — Wed, 08 Sep 2010 02:16:13 +0000

Apologies in advance for this being more meta-mathematics than mathematics. Reading over this approach to the EDP I am reminded of some notoriously thorny problems in operator theory: the Kadison-Singer problem, the Feichtinger conjecture, and other issues related to “pavable” operators and matrix pavings. Appropriately formulated, all concern the existence or nonexistence of various kinds of “decompositions” of matrices, with special attention paid to what is happening on the diagonal. (See e.g. http://www.aimath.org/WWN/kadisonsinger/ the paper “Equivalents of the Kadison-Singer problem.”) A good number of papers on pavable operators seem to follow the general theme that an operator that is “pavable” in any number of weak senses, must in fact be “pavable” in stronger, “more structured” senses. That seems very similar to what is desired here. (Nik Weaver even has a paper on Kadison-Singer stuff with the word “discrepancy” in the title— which is all I can say, not being all that familiar with his body of work, or any work on the EDP.)

Of course in this approach to the EDP it seems that the matrices one wants decompositions of are far from arbitrary (as they tend to be in Kadison-Singer type problems). They have a ton of extra “structure.” I wonder if there is anyone who is familiar with both this approach to the EDP and Kadison-Singer type problems and can comment on the distinction between the two. (e.g. are they similar enough that somebody with a strong opinion on the probable truth of the EDP should have a strong opinion on certain kinds of matrix pavings, or vice versa?)

By: Polymath5 « Euclidean Ramsey Theory

Polymath5 « Euclidean Ramsey Theory — Tue, 07 Sep 2010 19:03:18 +0000

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