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Jul 24, 2012 at 7:53 comment added Xander Faber Great answer! And now I see the mysterious $1 - 1/e$ showing up.
Jul 23, 2012 at 22:33 history edited Peter Mueller CC BY-SA 3.0
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Jul 23, 2012 at 18:35 comment added Seva Sounds interesting - will be happy to learn more about it. Keep in mind that results of this sort automatically yield an improvement in the finite field Kakeya problem (see the paper mentioned in my original post for the details).
Jul 23, 2012 at 17:34 comment added Xander Faber @Peter - Is your guess based on computation, or do you have a random model in mind?
Jul 23, 2012 at 15:13 history edited Peter Mueller CC BY-SA 3.0
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Jul 23, 2012 at 14:24 comment added Boris Bukh I get $83$ instead of $79$ from your example. It is still less than $2q/3$. No guarantees that my code is free of bugs though. Perhaps one can prove your hunch by taking a random function, and using a union bound together with a suitable deviation inequality.
Jul 23, 2012 at 13:45 history answered Peter Mueller CC BY-SA 3.0