Consider the set $A_N$ of all fractions $\left(\frac{3}{2}\right)^n \pmod{1}$ for $n\le N.$ Prove that $\min(A_N)→0$ as $N→∞.$

Question

This is related to the well-known unsolved problem in number theory that concerns the distribution of $(3/2)^n \pmod{1}$. This sequence is believed to be uniformly distributed. Has this simpler problem been proven before? I think that it may be done by a simple proof by contradiction, but my main concern is if it has been done before.

The set $A_7$ is $\{1/2, 1/4, 3/8, 1/16, 19/32, 25/64, 11/128\}.$ It seems very intuitive that the lower limit is 0.

To clarify, this question applies to n, where n is a natural number.

I had previously posted virtually an identical question, which got negative votes due to lack of clarity. However, I realized it is much easier to understand it in the (3/2)^n form. — user965964, Jul 07 '23 at 18:09
Re-posting is not allowed. As advised yesterday, you should edit your previous post instead, to improve it. And not only by reformulating the question: Quick beginner guide for asking a well-received question + please avoid "no clue" questions. — Anne Bauval, Jul 07 '23 at 18:51
https://math.meta.stackexchange.com/questions/2814/edit-and-rephrase-or-ask-new-question — Anne Bauval, Jul 07 '23 at 18:56

score -6 · Answer 1 · answered Jul 07 '23 at 18:30

-6

Since $A_0 = \{(3/2)^0 \text{ mod } 1 \} = \{1 \text{ mod } 1 \} = \{0\}$ and $A_0 \subset A_k$ for any $k$, we cleanly have $\min A_k=0$.

answered Jul 07 '23 at 18:30

Snared

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They clearly meant $n\geq 1.$ – MandelBroccoli Jul 07 '23 at 18:33
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I totally agree, but I also believe it is important to state one's question unambiguously. – Dan Asimov Jul 07 '23 at 18:41
@MandelBroccoli Where did they say to 1-index $A$? They specify all $n \le N_0$ in the question. not $1 \le n \le N_0$. Look a bit closer before downvoting next time - this is clearly the most concise solution – Snared Jul 07 '23 at 18:41
Also note I posted the answer before he edited his question to specify $A_7$ which implies 1-indexing, so I basically just got edit sniped – Snared Jul 07 '23 at 18:42
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@Snared it makes more sense to comment if their question is unclear or if they have seemingly missed a simple case – MandelBroccoli Jul 07 '23 at 18:45
@MandelBroccoli 3 days later and this answer is still correct yet -6 downvotes, do you see what you started? OP never said they wished to 1-index $A$ – Snared Jul 11 '23 at 22:45
I did mean to 1-index A or as I stated n is a natural number. I am really interested in knowing if this problem has been solved before. – user965964 Jul 14 '23 at 17:35
@user965964 well I solved your original question, I don't have time to check the new one – Snared Jul 14 '23 at 17:37
Typically you should post a new question rather than edit an old one into a new one after it has been answered - but anyway, hope you can figure out what you are trying to figure out – Snared Jul 14 '23 at 17:38

Consider the set $A_N$ of all fractions $\left(\frac{3}{2}\right)^n \pmod{1}$ for $n\le N.$ Prove that $\min(A_N)→0$ as $N→∞.$

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