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Oct
20
awarded  Yearling
Oct
6
comment Additive combinatorics and a Diophantine equation
You should be able to get something by combining a number of known results. After the BSG theorem you can apply a lemma of Ruzsa, which allows you to think of your sequence (or rather a suitable subsequence) as living inside a cyclic group that is not much bigger. Then there are results about long arithmetic progressions in sumsets. These will not necessarily be of the form (w,2w,3w,...) but I think a careful look at the proofs should give you this. See for example this paper: arxiv.org/pdf/1103.6000.pdf
Oct
4
awarded  Enlightened
Oct
4
awarded  Nice Answer
Oct
1
comment Proposals for polymath projects
This looks like a great question, and I agree that it would be good for a polymath project. Can I quickly check whether the rank you are talking about is over the reals? (I guess so, or you would have made it a 01 matrix.)
Aug
30
awarded  Good Answer
Aug
20
awarded  Great Question
Aug
7
awarded  Nice Answer
Aug
1
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May
21
awarded  Popular Question
May
15
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May
7
awarded  Notable Question
Apr
17
comment A variant of Goldbach Conjecture
Sorry, that wasn't what I meant to ask. I asked the question I actually wanted to ask in a comment on Harald's answer above, which he has now answered. (The question was whether one could do it for all N and not just sufficiently large N.)
Apr
16
comment A variant of Goldbach Conjecture
Can your work can be adapted to prove the result for every $N$ when $p_1,p_2$ and $p_3$ are required to be less than $N$?
Apr
15
comment A variant of Goldbach Conjecture
You mean the result where you insist that $p_1$, $p_2$ and $p_3$ are less than $N$?
Apr
10
awarded  Nice Answer
Apr
3
awarded  Famous Question
Mar
22
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Mar
20
awarded  Stellar Question
Mar
12
awarded  Good Answer