Wikipedia:Reference desk/Archives/Mathematics/2012 August 21

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August 21

Link to a result

I'm trying to find a proof online for a result from Hardy's book. Start with a fixed irrational number, say α, and consider the subset of the interval (0,1) given by N → (0,1), where n is sent to nα mod 1. The result states that the image of this function is dense in (0,1). — Fly by Night (talk) 17:48, 21 August 2012 (UTC)

I don't have a link to a proof, but I can give you a proof. Let x be our fixed irrational. For any integer q, put a(q) for the largest b so b / q < x. Then a(q) / q < x < (1 + a(q)) / q and since multiples of a given rational r cycle through {0/r,...(r - 1)/r} mod 1, we have for any b and q a y in the image so b/q < y < (1 + b)/q. So, given any two rationals s < t, pick b and q so s < b / q < (b + 1) / q < t; then the image must be dense since it has a member between every two rationals in the interval. Sorry if you were looking for some specific proof; also sorry for still not using the math style (esp. since you are the one who pointed it out to me!) :-) Phoenixia1177 (talk) 03:21, 22 August 2012 (UTC)

Incidentally, the answer to the previous question follows from this result (maybe that's why you brought it up?). Choosing the irrational number to be 1/2π, there are infinitely many integers n such that n/2π in an interval around 1/4 mod 1, so sin(n) will be near 1. Rckrone (talk) 03:47, 22 August 2012 (UTC)