Comments on: 254B, Notes 7: Sieving and expanders

By: Lew

Lew — Wed, 07 Mar 2012 19:20:39 +0000

The Landau conjecture mentioned (also attributed to Hardy & Littlewood) is possibly the “simplest” unsolved math problem, where “simplest” means shortest in first order number theory (using a suitable small set of symbols). See:

http://math.andrej.com/2006/11/04/are-small-sentences-of-peano-arithmetic-decidable/comment-page-1/

By: CE

CE — Tue, 06 Mar 2012 06:24:48 +0000

I think there is a typo: you mean: .
(An 11-almost prime has at most 11 prime factors, so a number with 6 prime factors (with multiplicity) is an 11-almost prime).

[Corrected, thanks - T.]

By: Terence Tao

Terence Tao — Tue, 06 Mar 2012 00:18:04 +0000

As stated in the main post, prime factors are being counted with multiplicity; thus, for instance, the number you mention is an r-almost prime for any .

By: petequinn

petequinn — Mon, 05 Mar 2012 20:08:23 +0000

Professor Tao, small question to clarify your definition of “almost primes.”

Is 2 x 2 x 2 x 3 x 3 x 5 a 3-almost prime or a 6-almost prime?

Thanks,

Pete

By: Terence Tao

Terence Tao — Mon, 05 Mar 2012 18:17:46 +0000

Fair enough; I edited the post accordingly. On the one hand, B-G-S already had the clean “Cartesian” approach to quasirandomness and non-concentration in their original paper, but the Cartesian approach to the product theorem (which is certainly the most difficult of the three ingredients to obtain) is clearly Varju’s contribution.

By: Ben Green

Ben Green — Mon, 05 Mar 2012 10:04:27 +0000

Terry, I think the the work of Varju was a more fundamental advance than you make out. The paper of Bourgain, Gamburd and Sarnak went into the inner workings of Helfgott’s paper and did everything with instead of . That was quite painful, because of the need for a satisfactory sum-product theory for . By contrast Varju took Helfgott’s conclusion for as a black box; his argument works for , for any semisimple . Nice post!

By: David Roberts

David Roberts — Mon, 05 Mar 2012 02:04:22 +0000

Small typo – definition of Fermat number is missing a 2 in the stack of exponentials: 2^n instead of 2^2^n.

[Corrected, thanks - T.]

By: Anonymous

Anonymous — Fri, 02 Mar 2012 19:48:26 +0000

Small typo: in the second paragraph, I think you want
\{ (n_1,n_2) \in \mathbf{Z}^2 : n_2 – n_1 = 2\}, not
\{ (n_1,n_2) \in \mathbf{Z} : n_2 – n_1 = 2\}

[Corrected, thanks - T.]

By: Jon

Jon — Fri, 02 Mar 2012 19:39:25 +0000

A very interesting topic, thank you for these notes!
An easy typo: in section 1, when introducing you correctly link to the primorial, but instead of using the word “primorial” in your sentence there’s a URL for the PNT (which perhaps you wanted to use in the previous paragraph, which mentions it without a link).

[Corrected, thanks - T.]