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Here's something I also posted on Stackexchange recently. It's very related to the Kakeya problem, yet I fail to see why this is true. It goes like this:

Let $r > 2$ be an integer parameter. Let $K \subset [N]=1,\ldots,N$ be a set containing arithmetic progressions of length $r$ in at least $N' \leq N/r$ (up to some constant) different dirrections ($b$ is the integer of the arithmetic progression $a,a+b,a+2b,\ldots$). Then, for every $r$, show that there exists a set $K$ as above of size at most $N^{1-\epsilon_{r}}$ with $\epsilon_{r} > 0$ a constant depending only on $r$.

I am kind of confused b/c I read some time ago about some result of J. Bourgain which basically said that a similar conclusion would give the validity of the general Kakeya, so I feel this is very hard, yet I got it as an exercise, so I'm probably missing something. Also, it would be very nice if someone can give some references about Bourgain's result which I'm talking about - I can't find it anywhere. Thanks

PS. can't login into the old account.

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Kakeya (for Minkowski dimension) would follow from the assertion that it is impossible to make $\epsilon_r$ a constant independent of $r$. –  Boris Bukh May 15 '12 at 10:42
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Someone has flagged this question, claiming that it is a homework problem. I don't know enough to tell if that is a reasonable conclusion. (What was your old account?) –  S. Carnahan May 15 '12 at 11:13
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Can you edit in the link to the stackexchange post? Wouldn't want to have any duplication of effort. –  Gerry Myerson May 15 '12 at 12:18

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