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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 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
comment 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
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 Frankl-Graham, 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 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
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 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.
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
Sep
20
awarded  Nice Answer
Jul
3
awarded  Great Answer