Kronecker theorem

Allnames

1)Prove that the sequence $\{na_n\}$ for all $n\ge 1$ is dense in $[0;1]$ if $a$ is an irrational humber .
2) Please give me some applications

**Posted:** Sun Jun 07, 2009 7:27 pm

MellowMelon

1) Suffices to show that we can find an $n$ for which $0 < \{ na \}$ is arbitrarily small, because we get all multiples of it. First, we show given $a_n$ there is an $n$ such that $\{na\} < \{a\}/2$ . Take the first $m$ for which $\{ma\} < \{(m-1)a\}$ (so we had a "carry"). Then $\{ma\} < \{a\}$ . If $\{ma\} < \{a\}/2$ we're done with $n = m$ . Otherwise as long as $\{(1+k(m-1))a\}$ is positive, $\{(1+(k-1)(m-1))a\} - \{(1+k(m-1))a\} = \{a\} - \{ma\} < \{a\}/2$ . Take the maximum $k$ such that $\{(1+k(m-1))a\}$ is positive, then that means $\{(1+k(m-1))a\} < \{a\} - \{ma\}$ and $n = 1+k(m-1)$ works. Now just replace $a$ with $na$ and repeat ad infinitum.

2) One of many: If $a$ is relatively prime to $10$ and $d_1d_2 \ldots d_k$ is a sequence of digits, there are infinitely many powers of $a$ that begin with $d_1d_2 \ldots d_k$ .

**Posted:** Sun Jun 07, 2009 8:14 pm

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Mashimaru

**Posted:** Wed Jul 29, 2009 9:37 am

tenniskidperson3

Apply the theorem to $\log a$ . Wink

**Posted:** Wed Jul 29, 2009 2:13 pm

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