Comments on: The sum-product problem http://ilaba.wordpress.com/2008/06/24/the-sum-product-problem/ Because "exact science is not always exact science." Tue, 03 Nov 2009 01:11:10 +0000 http://wordpress.com/ hourly 1 By: Izabella Laba http://ilaba.wordpress.com/2008/06/24/the-sum-product-problem/#comment-192 Izabella Laba Tue, 29 Jul 2008 17:53:52 +0000 http://ilaba.wordpress.com/?p=84#comment-192 Tim - You are right, it is indeed impossible for every element of $latex A/A$ to occur the same number of times, therefore one passes to a subset of $latex A/A$ (and the corresponding subset of pairs in $latex A\times A$) where each $latex a/a'$ occurs roughly the same number of times. Sorry about the imprecise wording. Tim – You are right, it is indeed impossible for every element of A/A to occur the same number of times, therefore one passes to a subset of A/A (and the corresponding subset of pairs in A\times A) where each a/a' occurs roughly the same number of times. Sorry about the imprecise wording.

]]> By: gowers http://ilaba.wordpress.com/2008/06/24/the-sum-product-problem/#comment-191 gowers Tue, 29 Jul 2008 16:45:48 +0000 http://ilaba.wordpress.com/?p=84#comment-191 Thinking about it further, I'm pretty sure that my guess about what you meant is indeed what you meant. Thinking about it further, I’m pretty sure that my guess about what you meant is indeed what you meant.

]]> By: gowers http://ilaba.wordpress.com/2008/06/24/the-sum-product-problem/#comment-190 gowers Tue, 29 Jul 2008 16:43:42 +0000 http://ilaba.wordpress.com/?p=84#comment-190 I'm sure I'm being stupid, but I don't see what you mean by reducing to the case where every element of A/A occurs the same number of times. Surely 1 will occur n times and max A/min A will occur once, whatever the set A is, so you can't get every element to occur even approximately the same number of times. Or do you mean that you keep A fixed but pass to a subset of A/A, losing at most a log factor, where each element occurs approximately the same number of times, and then argue somehow that "approximately" can be converted into "exactly"? Thanks for the post, by the way! I’m sure I’m being stupid, but I don’t see what you mean by reducing to the case where every element of A/A occurs the same number of times. Surely 1 will occur n times and max A/min A will occur once, whatever the set A is, so you can’t get every element to occur even approximately the same number of times. Or do you mean that you keep A fixed but pass to a subset of A/A, losing at most a log factor, where each element occurs approximately the same number of times, and then argue somehow that “approximately” can be converted into “exactly”?

Thanks for the post, by the way!

]]> By: Extermal Combinatorics II: Some Geometry and Number Theory « Combinatorics and more http://ilaba.wordpress.com/2008/06/24/the-sum-product-problem/#comment-179 Extermal Combinatorics II: Some Geometry and Number Theory « Combinatorics and more Thu, 17 Jul 2008 19:29:45 +0000 http://ilaba.wordpress.com/?p=84#comment-179 [...] at least . The exponent 5/4 was improved twice(!!) by Jozsef Solymosi and there is a very nice post about his most recent ingenious proof for a lower bound max (|A+A|, |A A|)  in Izabella [...] [...] at least . The exponent 5/4 was improved twice(!!) by Jozsef Solymosi and there is a very nice post about his most recent ingenious proof for a lower bound max (|A+A|, |A A|)  in Izabella [...]

]]> By: Gil Kalai http://ilaba.wordpress.com/2008/06/24/the-sum-product-problem/#comment-168 Gil Kalai Tue, 01 Jul 2008 15:28:49 +0000 http://ilaba.wordpress.com/?p=84#comment-168 Amazing! thanks.. Amazing! thanks..

]]> By: Anonymous http://ilaba.wordpress.com/2008/06/24/the-sum-product-problem/#comment-167 Anonymous Fri, 27 Jun 2008 01:17:18 +0000 http://ilaba.wordpress.com/?p=84#comment-167 Thanks for the exposition! I wonder whether at some point Jozsef (or somebody else) will come up with an equally simple and lucid argument for $latex \max\{|A+A|,|A\cdot A|\}>|A|^{3/2-\varepsilon}$? As far as typos are concerned - there is a minor one in the very first line (forgotten <EM>latex</EM>). Thanks for the exposition! I wonder whether at some point Jozsef (or somebody else) will come up with an equally simple and lucid argument for \max\{|A+A|,|A\cdot A|\}>|A|^{3/2-\varepsilon}?

As far as typos are concerned – there is a minor one in the very first line (forgotten latex).

]]> By: Izabella Laba http://ilaba.wordpress.com/2008/06/24/the-sum-product-problem/#comment-166 Izabella Laba Thu, 26 Jun 2008 05:31:19 +0000 http://ilaba.wordpress.com/?p=84#comment-166 D- Thanks for the correction! It's fixed now. D- Thanks for the correction! It’s fixed now.

]]> By: D http://ilaba.wordpress.com/2008/06/24/the-sum-product-problem/#comment-165 D Thu, 26 Jun 2008 05:21:00 +0000 http://ilaba.wordpress.com/?p=84#comment-165 I think things got scrambled here "In fact, if A is an arithmetic progression, then the sumset A+A is rather large, with |A+A|\approx |A|^2. Conversely, if A is a geometric progression, then A\cdot A must be large." I think things got scrambled here

“In fact, if A is an arithmetic progression, then the sumset A+A is rather large, with |A+A|\approx |A|^2. Conversely, if A is a geometric progression, then A\cdot A must be large.”

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