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.

Answer 1

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 $\min A_k=0$.