Ben Green
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 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. Nov 11 comment A question about groups of intermediate growth When making my comment above I was addressing the question for "supremum" instead of "infimum". I feel one does need Gromov/Pansu to do that. Nov 10 comment A question about groups of intermediate growth Nice question. My impression is that there is no simpler proof of 2. You can get some way beyond polynomial growth by quoting Shalom and Tao. Oct 25 comment Arithmetic Progressions of Squares Kevin - I'd suggest asking one of them. I think there is a preprint. Oct 25 awarded Enlightened Oct 24 awarded Nice Answer 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