3,558 reputation
12332
bio website maths.ox.ac.uk/~greenbj
location Auckland, New Zealand
age 38
visits member for 5 years, 1 month
seen yesterday

I'm a professor at Oxford University, on sabbatical in New Zealand until April 2014


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
Jun
25
awarded  Revival
Jun
12
comment 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
comment 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
comment 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
comment 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
comment 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