Is $\ \left\{\ \left(\frac{3}{2}\right)^n\mod 1:\quad n \in \mathbb{N}\ \right\}\ $ a dense subset of $\ [0,1]\ $?

Asked Aug 05 '21 at 15:11

Active Aug 05 '21 at 15:11

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Is $\ \left\{\ \left(\frac{3}{2}\right)^n\mod 1:\quad n \in \mathbb{N}\ \right\}\ $ a dense subset of $\ [0,1]\ $ ?

I know how to prove that if $\ \gamma\ $ is irrational, then sets like $\ \left\{\ \gamma\ n\mod 1:\quad n \in \mathbb{N}\ \right\}\ $ is a dense subset of $\ [0,1].\ $ But my question above seems unrelated to this, I think.

I've got a feeling the answer is yes, but I'm not sure how to go about proving it.

This question arose when thinking about this question I asked a few days ago.

asked Aug 05 '21 at 15:11

Adam Rubinson

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This is an open problem. – ə̷̶̸͇̘̜́̍͗̂̄︣͟ Aug 05 '21 at 15:26
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@TheSimpliFire feel free to post that as an answer. – Adam Rubinson Aug 05 '21 at 15:35
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"Well-known open problem" is a reason to close. – GEdgar Aug 05 '21 at 16:45
It is? I don't see why if there isn't a good dupe target. – Asinomás Aug 05 '21 at 16:52
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My lecture in Ergodic Theory mentioned in lectures that this is not known. – math111 Aug 05 '21 at 17:17

Is $\ \left\{\ \left(\frac{3}{2}\right)^n\mod 1:\quad n \in \mathbb{N}\ \right\}\ $ a dense subset of $\ [0,1]\ $?

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