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I would really appreciate some help with the following problem.

It resembles Van der Waerden a lot but I don't know how to proceed. I was told an averaging argument might do the trick but I can't see it.

Let $N, r$ be positive integers. Then, there exists a subset $X$ of $\left\{1,2,\ldots,N\right\}$ which contains arithmetic progression of length $r$ with at least $k \geq N/r \cdot \text{some constant}$ ratios (more precisely, it contains arithmetic progressions of length $r$ of the form $a,\ a+b,\ \ldots,\ a+rb$, where $b$ varies over $k$ values), and which is of size at most $N^{1-\epsilon(r)}$.

Thanks in advance!

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I would have thought that "translates" refers to the same thing with "$a$ varies" instead of "$b$ varies". These are more like scaled versions. Are you sure it's $b$ and not $a$? – joriki May 11 '12 at 6:22
The term "translates" made things unclear, yes. I edited; now things should be correct. – Anna May 11 '12 at 6:28
What does $k\ge O(N/r)$ mean? Usually $O$ is an upper bound (see e.g. here) and for instance the constant zero function is in $O(N/r)$, so that condition doesn't really tell us anything. Do you mean $k(N,r)\in\Theta(N/r)$? If so, that can't be true; all the $k$ values of $a+b$ are different, and if $k(N,r)\in\Theta(N/r)$, then that's already asymptotically more than $N^{1-\epsilon(r)}$ elements in the set. – joriki May 11 '12 at 6:44
@joriki: Perhaps what is meant by ratios is the ratios $\ell_1/\ell_2$ of terms $\ell_1,\ell_2\in X$? – anon May 11 '12 at 7:44
I abused the big $O$ notation; edited again – Anna May 11 '12 at 14:25

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