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May 6, 2019 at 19:09 comment added Joe Silverman There is a lower bound $H(f(\alpha))\ge CH(\alpha)^n$, but note that the implied constant will depend on $f$, as it will for the upper bound.
May 6, 2019 at 13:20 comment added Maurizio Moreschi The bound you prove can actually be improved to $H(f(\alpha))\le C H(\alpha)^n$ (which is what I was aiming for) under the assumption of my question. I post the proof of this as an answer, because it is too long for a comment.
May 3, 2019 at 2:26 history edited Bobby Grizzard CC BY-SA 4.0
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May 3, 2019 at 2:18 history answered Bobby Grizzard CC BY-SA 4.0