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bio website maths.ox.ac.uk/~greenbj
location Auckland, New Zealand
age 37
visits member for 4 years, 5 months
seen Sep 10 at 13:25

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


Oct
24
revised Arithmetic Progressions of Squares
added 270 characters in body; added 310 characters in body
Oct
24
answered Arithmetic Progressions of Squares
Oct
21
awarded  Enlightened
Oct
20
awarded  Nice Answer
Oct
20
comment Infimums of exponential sums involving primes
For a long time it was a conjecture of Littlewood that the infimum is $o(\sqrt{x})$. Actually it was Konyagin who got down to $x^{-1/2 + o(1)}$.
Oct
20
revised Infimums of exponential sums involving primes
added 385 characters in body; deleted 2 characters in body
Oct
20
answered Infimums of exponential sums involving primes
Oct
12
answered Cliques, Paley graphs and quadratic residues
Oct
4
comment density of a set
Why aren't they all dense? It's just a question of whether $(n^k/2\pi)$ is equidistributed modulo 1, and that will be so for any fixed k since $\pi$ is irrational. You'd use Weyl's inequality for a rigorous proof.
Sep
27
comment Extension of Tao-Green Theorem
I'm certain $A_1$ does contain long progressions, but proving it using the Tao-Green technique might not be so easy. In fact this set has density roughly $1/\log^2 N$ in the integers, and any such set is conjectured to have arbitrarily long progressions.
Sep
19
comment Elementary proof of the equidistribution theorem
This would basically be Weyl's proof, I think - the key idea there is to smooth the characteristic function of the interval [a,b] on which one wishes to count points n\alpha. The Fejer kernel (or de la Vallee Poussin kernel) is one way to do that. (I'd have to admit that I haven't bothered actually going to look at the book before making this comment.)
May
16
comment Size of Sum Sets
It doesn't prove anything, but the answer is indeed no, for essentially the reason you say. It can be proven rigorously using the circle method a la Vinogradov.
May
11
comment Lower bounds on the easier Waring problem
Boris, One would think so, because your set is contained in the difference set S of the set of things which are the sum of at most 5 positive kth powers, and there seems little reason to suspect that $S$ behaves so much unlike a random set with $n^{5/k}$ elements up to $n$. Proving it would be quite a different matter. Maybe some papers of Browning and Heath-Brown are relevant....
May
11
awarded  Nice Answer
May
7
answered Roth's theorem and Behrend's lower bound
Apr
24
awarded  Yearling
Apr
6
awarded  Enlightened
Apr
2
awarded  Nice Question
Mar
31
comment Can Hölder's Inequality be strengthened for smooth functions?
Hmm, it's not so clear. The problem is that it seems to be hard to say much about f. For example, f could consist of spikes of height $N$ and width $1/N^2$ about a Sidon set of size about $N$; then $f \ast f$ will look a bit like the characteristic function of a union of $N$ intervals of width $\sim 1/N$; call this set $E$. Unfortunately I can't conclude anything useful about $\Vert \hat{1}_E \Vert_1 = \Vert f \Vert_2^2$ being small.
Mar
31
comment Can Hölder's Inequality be strengthened for smooth functions?
Terry, instead of using the idempotent theorem I think a result of Rudin, improved by Saeki is more relevant: MR0225102 (37 #697) Saeki, Sadahiro On norms of idempotent measures. Proc. Amer. Math. Soc. 19 1968 600–602. In R, where there are no interesting compact subgroups, this will tell you that $\|\hat{1}_E\| > 1.2$. WIll have to think about the details.