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Apr
28
answered Arbitrarily many primes in a Fibonacci-type 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 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