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Jul 28, 2014 at 19:33 comment added sb945 Johnston's comment is of correct understanding, I re-organized my question to make it clearer
Jul 28, 2014 at 13:41 comment added Nathaniel Johnston The question is asking for the best possible upper bound in terms of r, so what is needed is either an example that gets the ratio much larger than 1 (i.e., close to $\sqrt{r}$) or a proof of a better upper bound (e.g., something like $\log(r)$).
Jul 28, 2014 at 5:23 history answered Mustafa Said CC BY-SA 3.0