3,403 reputation
12132
bio website maths.ox.ac.uk/~greenbj
location Auckland, New Zealand
age 37
visits member for 4 years, 4 months
seen Jan 31 at 3:07

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


Sep
16
comment There exists B subset A, |B| = log n, A \cap 2*B = \emptyset
A masters student from Lyon, Jehanne Dousse, recently improved [SSV] to log n logloglog n or thereabouts, by replacing the use of Szemeredi's theorem by a density increment argument generalising that of Roth, allowing one to locate inside a dense set A some elements x_1,...,x_k, all of whose midpoints also lie in A. The key is that these configurations can still be detected by Gowers' U2-norm. I have no clue what the correct bound is....
Jul
28
comment Growth rate of the infinity norm of Discrete Fourier Transform of +1,-1 vectors
The answer is basically $\sqrt{n}$ for large $n$.When $n$ is prime, an example should be given by the Legendre symbol $f_i = (i | n)$.
Jun
18
awarded  Enlightened
Jun
18
awarded  Nice Answer
Jun
18
answered Can you cover the Boolean cube {0,1}^n with O(1) Hamming-balls each of radius n/2-c*sqrt(n)?
Jun
18
comment Can you cover the Boolean cube {0,1}^n with O(1) Hamming-balls each of radius n/2-c*sqrt(n)?
I agree it should probably be linear in n, but this might be rather hard to prove. Benny Sudakov was telling me that it is known (but difficult) that Hamming Balls of radius 1 cover with efficiency 1 + o(1), but this is an open problem for balls of radius 2. He may known about your question too; I'll ask him this afternoon.
May
22
comment Nonlinear equations in integers
In fact there is no set of size $o(N)$ which misses all triples $(3k, 4k, 5k)$ by the following type of argument: consider such triples with $|k - N/10| < N/100$ (say). These are all disjoint. Therefore any set consisting of $98\%$ of ${1,..,N}$ contains a pythagorian triple. Computation of the exact density might be tricky.
May
22
comment Nonlinear equations in integers
One of the references here, I bet: algo.inria.fr/csolve/triple
May
22
comment Nonlinear equations in integers
I'm pretty sure, actually, that all sets of $99\%$ of $[1,...,N]$ contain a triple $(3k, 4k, 5k)$. I'm also pretty sure I can find a reference for this fact given a minute or two.
May
22
comment Nonlinear equations in integers
Mark, for the Pell equation there is a huge set $A$, of size $N - O(\log N)$, just by deleting all $x$ and $y$ that are solutions to the Pell equation?
May
22
comment Nonlinear equations in integers
Quid: what about $3^2 + 4^2 = 5^2$?
May
22
comment Nonlinear equations in integers
Siming, I think that taking the squares $x^2$ with $x$ either odd or $\equiv 2 \mod{4}$ gives you a set consisting of $\frac{3}{4}$ of the squares with no solution to $x^2 + y^2 = z^2$ (look mod $8$). Probably it's true that if you take $99 \%$ of the squares then there's a solution to $x^2 + y^2 = z^2$. I don't immediately have a feel for whether this is doable or not; I'll get back to you. Certainly use of the circle method will be problematic if one proceeds naively.
May
22
answered Nonlinear equations in integers
May
17
comment Gauss sums over multiplicative subgroups
Igor, Thanks for this mention. I did try pretty hard with that particular set of notes, though I might change one or two things if I were doing them again. I had some help from Bourgain and Lindenstrauss.
Apr
23
awarded  Yearling
Apr
12
comment Why groups that admit Folner Sequences are amenable
Jo, you can use the Folner sequence to define an almost invariant mean, then take a limit of these along an ultrafilter to get a genuinely invariant one. See for example these notes, starting page 26, for a discussion in the case of Z. dpmms.cam.ac.uk/~bjg23/ATG/Chapter3.pdf I'm speaking here of the case when G is a discrete group; in the locally compact case matters are a little more complicated. There are many better sources in the literature - recent blog notes of Tao, to give just one example.
Apr
8
awarded  Necromancer
Apr
2
revised Perron, Fourier
deleted 1 characters in body
Apr
1
awarded  Nice Answer
Mar
28
answered Perron, Fourier