bio | website | maths.ox.ac.uk/~greenbj |
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location | Auckland, New Zealand | |
age | 38 | |
visits | member for | 5 years, 1 month |
seen | yesterday | |
stats | profile views | 12,509 |
I'm a professor at Oxford University, on sabbatical in New Zealand until April 2014
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
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Apr 1 |
awarded | Nice Answer |
Mar 28 |
answered | Perron, Fourier |
Mar 19 |
comment |
About unpublished lecture notes of Philip Hall
There's a copy of the Edmonton notes in the Cambridge maths library; I consulted it a couple of years ago. |
Feb 3 |
revised |
minimum number of subsets?
added 392 characters in body; added 131 characters in body |
Feb 3 |
answered | minimum number of subsets? |
Dec 27 |
comment |
Every prime number > 19 divides one plus the product of two smaller primes?
Denis, I don't understand your comment. I only know two things that count as good reasons in the theory of prime numbers. First, a proof. Second, a decent heuristic or "statistical" argument. Here, the set of $1 + rs$ ought to look like a fairly random (give or take some irregularities mod small primes) subset of $[1,p^2]$ of density $1/\log^2 p$. The probability of $x \leq p^2$ being divisible by $p$ is $1/p$. I'd expect subsets of $[1,X]$ of densities $\alpha$ and $\beta$ to intersect as soon as $\alpha \beta \gg 1/X$ unless there is some good reason why not. |
Dec 27 |
comment |
Every prime number > 19 divides one plus the product of two smaller primes?
Geoff- there certainly is work on this. It's known that the smallest prime congruent to $a$ mod $q$ is $\ll q^{5.4}$ (or so). It's conjectured that it's $\ll q^{1 + \epsilon}$. The GRH would give $\ll q^{2 + \epsilon}$. |
Dec 27 |
comment |
Every prime number > 19 divides one plus the product of two smaller primes?
Mark - yes, there are "sum-product" results of this type. The one most applicable here is that due to Bourgain-Katz-Tao: it says that if $A \subseteq Z/pZ$ and $|A + A|, |A \cdot A| \leq K|A|$ then either $|A| \leq K^C$ or $|A| \geq K^{-C} p$. When $A$ has size about $p/\log p$, it tells you rather little (namely either $|A + A|$ or $|A \cdot A|$ has size at least $p/(\log p)^{1 - c}$) and I think exponential sum techniques are likely to be better. |
Dec 27 |
comment |
Every prime number > 19 divides one plus the product of two smaller primes?
I might add that I would expect trigonometric sums to allow one to show that $A \cdot A$ is almost all of $Z/pZ$ as $p \rightarrow \infty$. |
Dec 27 |
comment |
Every prime number > 19 divides one plus the product of two smaller primes?
Interesting problem. I don't think either the sum-product idea or the trig sums idea will work immediately. In the latter case because one has a binary problem instead of one with three variables, and in the former because I think you can have both $A.A \subsetneq Z/pZ^*$ and $A + A^{-1} \subsetneq Z/pZ$ with $A$ reasonably large, even positive density. But both of these are good ideas on which to base further thought. |
Dec 19 |
awarded | Notable Question |
Dec 16 |
comment |
Picking out an odd subset
I should also add that the second question is rather natural: if I tell you $M(p,h)$ for all $h$ mod $p$ and for all $p \leq X$, how big does $X$ need to be before you can tell me $|M|$? Perhaps I'll ask some of the Bristol number theorists today. |
Dec 16 |
comment |
Picking out an odd subset
Perhaps ideas connected with large sieve will help here. If $M(p,h)$ denotes the number of elts of $M$ congruent to $h$ mod $p$ then the assumption that $M(p,h)$ is always even should force the square mean $\sum_h |M(p,h) - |M|/p|^2$ to be, say, twice the size that one might expect for a positive fraction of $p$s. I'm not claiming to have an actual argument, though! |
Dec 12 |
answered | Is there a Plancherel Theorem for Gowers norms? |
Dec 2 |
comment |
A conjecture on the relative size of Goldbach pairs?
There is most likely a literature on results of the form "For almost all even numbers 2N we can write 2N = p_1 + p_2 with p_1/p_2 < 1/F(N)". I'm pretty sure simply getting F(N) -> infty would be a simple application of the circle method, but I'd be surprised if one could get much better than F(N) > N^c for some small c, if that. |
Nov 26 |
comment |
Minimum number of twin primes < N
Greg, I was tempted to quibble this comment. But then it occurred to me that there really aren't any modern approaches to the twin prime conjecture. One could imagine an approach using nonstandard analysis, though, which would give no quantitative lower bounds. |