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Jun
10 |
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Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture
I may try and read BFI. If you do it just in some critical range, rather than in the extreme generality they do with unspecified parameters $C, D, H, R, N,\dots$, my guess is it will boil down to the same basic ingredients in a different order, once one has applied a suitable decomposition into bilinear forms: Cauchy, Weyl shift, completion of sums/Fourier expansion. What else is there? |
Jun
9 |
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Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture
And by the way, I think he should have called his paper "On a new bound for an incomplete Kloosterman sum to composite moduli, with applications". |
Jun
9 |
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Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture
I know very little of Kloostermania, but my impression is the arguments in the BFI papers you speak of are basically the same, only the inputs are bounds for averages of Kloosterman sums over different moduli (?) coming from automorphic form theory, rather than bounds for products of Kloosterman sums to a distinct modulus. So it's just a different black box you have to believe, right? |
Jun
9 |
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Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture
I have to say that, now that I have studied the paper in some detail, this seems to me to be an extraordinarily accurate (if perhaps difficult to parse on a first reading) answer to the original question. |
Jun
1 |
awarded | Good Answer |
May
24 |
awarded | Nice Answer |
May
21 |
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Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture
I'll denote any points I get from this observation (which I stole from someone else anyway) to a good cause. An additional point to be made, now the preprint is available, is that the Bombieri-Fouvry-Friedlander-Iwaniec type results on level of distribution depend on an estimate for sums of Kloosterman sums that requires (via a lemma of Bombieri and Birch) Deligne's work on the Weil Conjectures. It seems to me (though I'm certainly not an expert) that these estimates are not accessible by the more elementary methods such as Dwork/Stepanov. |
May
21 |
answered | Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture |
May
20 |
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Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture
Mark - we added our comments at the same time. I was referring to the table on page 12 of GPY, whereas you were referring to the one on p9. I guess the one on page 9 tells you what you can get using just the basic GPY method, and the one on page 12 uses more complicated weights. I guess Zhang elaborates on the basic GPY method. Maybe his method can be combined with the more complicated GPY method which leads to the numerics on page 12; it seems likely that this will be one place any would-be 70000000-reducers will look first. |
May
20 |
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Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture
It's interesting to speculate on how much the 70000000 will be reduced. In this regard, the table on page 12 of Goldston-Pintz-Yildirim is relevant. If one had level of distribution 4/7 with no strings attached, it seems one might get gaps of size 500 or so. To get gaps under 100 without some completely new idea one would have to go out to level of distribution nearly 2/3, i.e. double the improvement of BFI. I'd wager that getting down to 10000 or so is going to prove pretty difficult. |
May
15 |
awarded | Nice Answer |
May
14 |
answered | Proof of the weak Goldbach Conjecture |
Apr
23 |
awarded | Yearling |
Apr
19 |
accepted | Additive Combinatorics - reference request |
Apr
19 |
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Additive Combinatorics - reference request
I finally dug out the reference myself using a MathSciNet search. MR1931192 (2003f:11016) Reviewed Schoen, Tomasz(PL-POZN) The cardinality of restricted sumsets. (English summary) J. Number Theory 96 (2002), no. 1, 48–54. |
Apr
19 |
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Additive Combinatorics - reference request
Thanks Seva - but I'm sure I've seen this exact argument somewhere. I guess not in one of your papers then? |
Apr
19 |
asked | Additive Combinatorics - reference request |
Apr
9 |
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Minimal size of subsets $A,B$ in a finite group $G$ such that $AB=G$
Is this exactly "coupon collecting analysis"? For each choice of B = {b_1,..,b_k} you can try and estimate the probability that AB is all of G, if A is chosen randomly. But then you have to sum over a large selection of B's, and the triangle inequality may be too crude. In fact, I posed as an open problem (Barbados workshop, March 2012) the question of deciding whether the log N should really be there. Can Z/NZ be covered by O(N^{1/2}) translates of a random set A of size N^{1/2}? |
Apr
8 |
awarded | Good Answer |
Mar
26 |
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Riemann Z function, bounds on number of non-trivial zeros along horizontal lines, rather than vertical ones
I think that from standard asymptotics on the number of zeros of height up to T you get a bound of O(log T). Apparently even on RH is is only known that the multiplicity of 1/2 + iT as a zero is O(log T/log log T). So I guess you shouldn't expect too much better than this O(log T) bound, |