Irrational number and Dense set

1001001 · New Member Offline Joined: 17 Jun 2007 Posts: 8

If $a$ is a irrational number then set $A =$ { $a2^n - [a2^n],n \in N$ } is dense in $[0,1]$ .
Prove or disprove it.

**Posted:** Thu Dec 20, 2007 7:58 pm

pleurestique

false. if, say, $a = 0.101001000100001...$ in base 2, then there are no fractional parts of $2^na$ in an interval around 1.

**Posted:** Thu Dec 20, 2007 9:15 pm

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