This is a test message. Maths content to follow, but I'll wait until more people sign up. Best, Terry -- Terence Tao, Department of Mathematics, UCLA ...
Forwarded message from Josh: There is an interesting result in section 3.1 of the paper "Optimization in arrangements" by Langerman and Steiger (attached). In ...
So I think I can get the first non-trivial result about point-plane incidences - well short of full generality, but a step in the right direction: Proposition....
Great, Terry! Was lying in bed thinking about this very question this morning. I tend to agree with what you say here. I will try to dig up and or work out...
From Nets's e-mail: "Jozsef really wants to work on higher d Erdos, thinks his product trick may simplify some of our incidence theory, and also points out...
... Jordan, In 3d, the triangle problem is: Given N points in R^3, show that these points define at least N^2 non congruent triangles. If they do, there must...
So it looks like we understand pretty well now how to do incidences between points and lines in any dimension (modulo Claim 1 and Claim 2, which look close to...
Sorry, should have clarified. Traditionally, k is the multiplicity; the average number of lines/planes/etc. that are incident to each point (or higher dim...
I started writing up a proof of Claim 3 and then realized that there are perhaps some actual subtleties to be dealt with. For example: suppose S is a set of...
Hmm. In that _specific_ case, we can throw away half of the points and end up in a much better situation... more generally, if we can just get a sizeable...
Oh, that's good news (that we only need a small fraction of the n.) In that case it should work something like this. You choose P_1 to have degree as low as...
I think for now we should be happy to lose as many logs as is needed to get a working theorem. After that we can try killing off the logs one by one, though I...
OK, I think I now have the right numerology to solve the d-dimensional Erdos problem using the triangle formulation rather than the double coset formulation;...
Hi, Following Nets' instructions I joined to the group. In Elekes' program for me the great difficulty was to prove the k=2 case. That is, to show that the...
For point-plane incidences there is a sharp bound due to Elekes and Toth: Gy. Elekes, Cs.D. Tóth: Incidences of not too degenerate hyperplanes. You can...
The two-dimensional argument should work in higher dimensions as well if the k=d case could be done. Let me state the incidence problem first. Two sets, A and...
Dear Jozsef, The numerology (reducing things to the k=O(1) case) seems to be good; I was working with a slightly different setup (counting rigid motions...
Thanks Terry, So, it seems that the bottleneck - unless we overlooked something - is the k=d case. Can we formulate it as an unrestricted incidence problem?...
Well, I'm thinking of it as an incidence problem in the d(d+1)/2-dimensional space SE(d) of rigid motions on R^d, between N^{d+1} "points" (i.e. rigid motions)...
Just a note -- I am just trying to write down the right form of the "multiple polynomial method" -- I am pretty optimistic that one can write something down...
Just a note: the following paper of Atsushi Ikeda just posted to arXiv should be a good up-to-date reference for some of the issues surrounding "Claims 1 and...
I've uploaded the file polyham_for_lines.pdf, which contains a sketch of an argument showing that if we have m families of lines in R^n, we can find a...
Hi guys, I just uploaded a short .pdf about the "double polynomial method." What's proved there, if I did it right, is that if you have N points in P^n (or...
Nice! What's tantalising is that we are so close to the dimensions we need to do the 3D erdos problem, which concerns incidences of 3D objects in R^6. So ...
Hi everyone, I wrote some notes on point-plane incidences in R^n and put it on the yahoogroup. From looking at the grid example, one expects that given ...
I think/hope that the requirement of complete intersection can be productively weakened, but it's on the non-algebraic side that I get confused about this....
Will read this on plane tomorrow, Terry, looking forward! But one comment on Erdos 3-D following on from my last e-mail. Let G be the group of rigid motions of...
Indeed, here's another fact one might hope makes life easier. In planes.pdf, where you try to control incidences between points and planes, you have to...
Dear Jordan, You've convinced me that we should be attacking the 3D Erdos problem directly now, because the toy problems of 2-planes in R^n etc. are not ...
