Timeline for 1 rectangle <= 4 squares
Current License: CC BY-SA 2.5
43 events
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| Mar 24, 2010 at 19:19 | comment | added | Yaakov Baruch | I think I was wrong in the main question where I stated that b(ZxZ)<=254/67. What is true is that if f is bounded on rectangles, then f(any rectangle)<=254/67. But I don't see a way to eliminate the possibility of a runaway f (i.e. unbounded on rectangles). I'll correct the main question if/when I have something else of substance to add. | |
| Mar 12, 2010 at 18:59 | comment | added | Yaakov Baruch | It seems that B(Z/2 x Z/2)=3, but I need to check my calculations (basically the awk program provided in one of the older answers below, modified to do arithmetic mod 2). | |
| Mar 12, 2010 at 11:47 | comment | added | Yaakov Baruch | B(Z/2)=B(Z/3)=1, B(Z/4)=B(Z/5)=2. | |
| Mar 12, 2010 at 4:40 | comment | added | Yaakov Baruch | The group theoretic version could be formulated this way too: given an abelian group G and all maps f: GxG->R such that -1<=f(c,d)-f(a,d)-f(c,b)+f(a,b)<=1 for all a,b,c,d in G such that ad=bc, then find the a best limit B(G) such that -B(G)<=f(c,d)-f(a,d)-f(c,b)+f(a,b)<=B(G) for all a,b,c,d in G. It is easy to prove that B(G/H)<=B(G) for any group G and subgroup H, by lifting any example f: G/HxG/H->R to an example F:GxG->R via the canonical projection. | |
| Mar 10, 2010 at 16:40 | history | edited | Yaakov Baruch | CC BY-SA 2.5 |
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| Mar 10, 2010 at 16:33 | history | edited | Yaakov Baruch | CC BY-SA 2.5 |
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| Mar 9, 2010 at 1:27 | history | edited | Yaakov Baruch | CC BY-SA 2.5 |
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| Mar 9, 2010 at 1:20 | answer | added | Yaakov Baruch | timeline score: 3 | |
| Mar 7, 2010 at 23:18 | history | edited | Nurdin Takenov | CC BY-SA 2.5 |
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| Mar 7, 2010 at 22:36 | history | edited | TonyK | CC BY-SA 2.5 |
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| Mar 5, 2010 at 23:32 | comment | added | F_G | I have setup a Google Docs page to record progress on this problem, hopefully we can stop continually updating this page. It is at docs.google.com/View?id=ajkfbpjb4hfn_182c4hq2qfj If you want to help edit the page, send me an email ; by the way, for those following the search for better configurations, there is something new there (a new 1x12 solution ...) | |
| Mar 5, 2010 at 9:37 | comment | added | Yaakov Baruch | This whole page is becoming messy... I have a comment to one of Tony's answers below about the 3.8 limit, but perhaps should we all agree to post future comments only to the main question or the most recent answer? (We should try to limit the use of updates to significant developments only, lest other users object to this question getting bumped to first page too often.) | |
| Mar 4, 2010 at 11:25 | history | edited | TonyK | CC BY-SA 2.5 |
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| Mar 2, 2010 at 20:53 | history | edited | Yaakov Baruch | CC BY-SA 2.5 |
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| Mar 2, 2010 at 20:18 | answer | added | Yaakov Baruch | timeline score: 4 | |
| Mar 2, 2010 at 20:15 | answer | added | F_G | timeline score: 5 | |
| Mar 2, 2010 at 13:11 | answer | added | TonyK | timeline score: 1 | |
| Mar 2, 2010 at 7:20 | history | edited | Yaakov Baruch | CC BY-SA 2.5 |
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| Mar 2, 2010 at 7:13 | history | edited | Yaakov Baruch | CC BY-SA 2.5 |
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| Mar 1, 2010 at 23:10 | answer | added | TonyK | timeline score: 10 | |
| Feb 26, 2010 at 2:36 | answer | added | Yaakov Baruch | timeline score: 10 | |
| Feb 9, 2010 at 12:22 | comment | added | Yaakov Baruch | would it make sense to ask the following in R^2 for C^-1 functions: if the integral for a function f is limited between +-1 on each disk, is there a limit on every convex set? | |
| Feb 7, 2010 at 20:17 | answer | added | F_G | timeline score: 13 | |
| Feb 5, 2010 at 11:59 | history | edited | Yaakov Baruch | CC BY-SA 2.5 |
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| Feb 4, 2010 at 11:17 | history | edited | Yaakov Baruch | CC BY-SA 2.5 |
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| Feb 4, 2010 at 10:58 | history | edited | Yaakov Baruch | CC BY-SA 2.5 |
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| Feb 3, 2010 at 15:07 | comment | added | Douglas Zare | I retagged to combinatorics and created a discrepancy-theory tag. It's possible that nt.number-theory would fit, since this is within combinatorial number theory and some discrepancy theory papers get tagged nt, but I think that's misleading. Although this is "puzzling" and can be described using "integrals" over "measures," I don't think those fit as tags. | |
| Feb 3, 2010 at 15:03 | history | edited | Douglas Zare |
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| Feb 3, 2010 at 13:44 | answer | added | domotorp | timeline score: 6 | |
| Feb 3, 2010 at 11:12 | comment | added | Yaakov Baruch | I agree it's questionable tag, unless perhaps we thing in terms of some generalized theory that allows negative measures (is there such?). I added "integration" and "inequalities" and I'm open to any suggestion, including removing "measure-theory". | |
| Feb 3, 2010 at 11:10 | history | edited | Yaakov Baruch |
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| Feb 3, 2010 at 7:09 | comment | added | Yemon Choi | Interesting problem/result. One question: why the "measure-theory" tag? | |
| Feb 3, 2010 at 0:36 | comment | added | Douglas Zare | Why not give the proofs for 4 and 10/3? | |
| Feb 2, 2010 at 17:22 | comment | added | Yaakov Baruch | or more likely Bercovici | |
| Feb 2, 2010 at 17:20 | comment | added | Yaakov Baruch | Hari Berkovici. | |
| Feb 2, 2010 at 16:25 | comment | added | Kevin O'Bryant | Professor's name? | |
| Feb 2, 2010 at 13:39 | comment | added | Yaakov Baruch | Kevin, your restatement is correct. I wrote the question the way I did because the proof I know is geometric. | |
| Feb 2, 2010 at 13:37 | history | edited | Yaakov Baruch | CC BY-SA 2.5 |
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| Feb 2, 2010 at 13:26 | comment | added | Kevin Buzzard | I still found the question hard to understand. Lemme just check: is it the following? given f:Z^2->R with, for all integers i,j and N>=0, |sum_{i<=x<=i+N,j<=y<=j+N}f(x,y)|<=1, then for all i,j and M,N>=0, |sum_{i<=x<=i+N,j<=y<=j+M}f(x,y)|<=4? I'm not saying you should rewrite it like this, I just don't have a very geometric mind. | |
| Feb 2, 2010 at 12:55 | history | edited | Yaakov Baruch | CC BY-SA 2.5 |
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| Feb 2, 2010 at 12:31 | history | edited | Yaakov Baruch | CC BY-SA 2.5 |
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| Feb 2, 2010 at 12:26 | comment | added | TonyK | Are your squares and rectangles parallel to the coordinate axes? Does your sum include all points on or inside the square/rectangle, or just the points on the boundary (or just the corners)? | |
| Feb 2, 2010 at 12:14 | history | asked | Yaakov Baruch | CC BY-SA 2.5 |