Ben Green
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 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 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?