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Jacobi's

Kronecker's theorem asserts that if $\lambda$ is irrational then the orbit of $n\lambda$ for $n=1,2,3,\ldots$ is dense in $S^1$ $\simeq$ $\mathbb{R}/\mathbb{Z}$. A proof uses the Pigeon-hole Principle. It relies on the fact that if you divide $S^1$ into $k$ equal (but very small) segments you must hit one of these segments twice, by the Pigeon-hole Principle.

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Jacobi's theorem asserts that if $\lambda$ is irrational then the orbit of $n\lambda$ for $n=1,2,3,\ldots$ is dense in . $S^1$ $\simeq$ $\mathbb{R}/\mathbb{Z}$. A proof uses the Pigeon-hole Principle. It relies on the fact that if you divide $S^1$ into k $k$ equal (but very small) segments you must hit one of these segments twice, by the Pigeon-hole Principle.

1 [made Community Wiki]

Jacobi's theorem asserts that if is irrational then the orbit of for is dense in . A proof uses the Pigeon-hole Principle. It relies on the fact that if you divide into k equal (but very small) segments you must hit one of these segments twice, by the Pigeon-hole Principle.