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answered  Arbitrarily many primes in a Fibonaccitype sequence 
Apr
23 
awarded  Yearling 
Dec
21 
comment 
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/… 
Dec
1 
awarded  Notable Question 
Nov
18 
comment 
A characterization of quadratics similar to an inverse sieve problem
Here is the paper. arxiv.org/abs/math/0304183 
Nov
17 
comment 
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 deBruijn 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 Fourierbased 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 wellknown 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 
comment 
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 FranklGraham, which give many monochromatic solutions). Contradiction. 
Jul
26 
comment 
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 
comment 
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 CauchySchwarz 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 
comment 
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. 
Apr
23 
awarded  Yearling 
Mar
19 
comment 
Cricket and the HardyLittlewod 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. 
Jan
30 
awarded  Pundit 
Aug
26 
comment 
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. 
Apr
23 
awarded  Yearling 
Feb
18 
awarded  Enlightened 
Feb
18 
awarded  Nice Answer 
Jan
12 
awarded  Autobiographer 
Jan
12 
awarded  Disciplined 