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Dec
21 |
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The original proof of Szemerédi's Theorem
www.math.ucla.edu/~tao/preprints/Expository/szemeredi_theorem.dvi : though Tao later states that in this exposition he "was unable to find much of a simplification, and my exposition is probably not that much clearer than the original text". See also terrytao.wordpress.com/2012/03/23/… |

Nov
18 |
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A characterization of quadratics similar to an inverse sieve problem
Here is the paper. arxiv.org/abs/math/0304183 |

Nov
17 |
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A characterization of quadratics similar to an inverse sieve problem
This is a pretty question, and it could be hard. I would phrase it as follows: let $B \subset \{1,\dots, X\}$ be the set of $10\sqrt{X}$-smooth numbers, so $|B| \sim c X$ for a certain $c$ given by the Dickman de-Bruijn function. What is the largest $S$ for which $S - S \subset B$? In general it's very, very hard to show that $|S| < \sqrt{X}$ in problems of this type, because Fourier-based methods don't work. For a generic $B$, one expects the biggest $S$ to have size around $O(\log X)$ (in fact I proved this) so to have such a large $S$ indicates an unusual property of this particular $B$. |

Nov
17 |
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Is there a “complete” Sidon sequence?
No, there is a well-known result of Erdos (mentioned on the Wikipedia page for Sidon Sequences) saying that an infinite Sidon sequence must have liminf a_n/n^2 = 0, whereas a sequence with the property you mention cannot have this property. I believe you can find the proof in Halberstam and Richert's sequences book, though I do not currently have it to hand. |

Nov
4 |
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Partition regularity in the squares
My guess is that the answer to your Question 2 should be "yes" using the transference principle as articulated by Browning and Prendiville. This will let you transfer a putative colouring of the squares with no solution to c_1 x_1 + ... + c_s x_s = 0 (with the x_i squares) to a colouring of all of {1,..,N} with few solutions to the same equation, provided s >= 5. But then choose some c_i which do not sum to zero, and for which Rado's theorem holds (more accurately the quantitative strengthenings of Rado due to Frankl-Graham, which give many monochromatic solutions). Contradiction. |

Jul
26 |
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Which journals publish applied mathematics with mostly pure mathematics content?
Math Proc Camb Phil Soc: journals.cambridge.org/action/displayJournal?jid=PSP (If it doesn't satisfy point 1 of your requirements then I don't think I want to know :) ) |

May
2 |
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Largest Fourier coefficient of sparse boolean function
Tom got back to me. You can indeed get such a bound by considering the (2k)th moment of f^(gamma), for which you can find a lower bound by expanding out combinatorially and using the Cauchy-Schwarz inequality to bound the number of solutions to x_1 + ... + x_k = y_1 + ... + y_k with f(x_i), f(y_j) = 1. Then ignore the term gamma = 0 and optimise by taking k ~ n/c. |

May
2 |
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Largest Fourier coefficient of sparse boolean function
I think Tom Sanders considered this question and may have had an argument to show that F(n,cn) > c' 2^n, where c' depends only on c, but I don't remember exactly. I'll ask him. I think this is closely related to the issue of getting a bound on the Sidon constant s(X \cup Y) in terms of s(X) and s(Y) (known, due to work of Rider in the 1970s), together with the fact that a dissociated subset of GF(2)^n is Sidon, where Sidon is in the harmonic analysis sense webpages.uidaho.edu/lnguyen/SidonSet_RieszProduct.pdf. |

Mar
19 |
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Cricket and the Hardy-Littlewod maximal function
From what I understand the theorem is rather trivial when applied to Bollobas's own cricketing career - run out for 0 off the second ball of his only innings. |

Aug
26 |
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Maximum sets of lattice points such that only a few points collinear
It is, in fact, a very unsolved problem to decide whether you can put $2n$ points in $[n] \times [n]$ with no three collinear, if $n$ is large. I suspect the answer is no, and that in fact one can only put $(c + o(1))n$ such points for some $c < 2$; possibly $c = 3/2$. There is a construction which achieves this. |

Jun
12 |
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What are some examples of mathematicians who had an unconventional education?
The windmill in Nottingham operated by Green (no relation) can still be visited: en.wikipedia.org/wiki/Green's_Mill,_Sneinton though it is something of a detour even if one happens to be in Nottingham. Isaac Newton's home, 28 miles away, could be ticked off too for a mathematical tour of the East Midlands of England. |

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. |

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
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. |

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? |