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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? |