query: If $R:S^{1}\rightarrow S^{1}$ is a irrational rotation, $\{R^{n}([x])\}$ is dense in $S^{1}$ for all points.
positive_passages.docid: 142577
positive_passages.text: Algebraic proof that a set generated by irrational rotations is dense in $S^1$.
negative_passages.docid: 10008
negative_passages.text: Proof that the nth root of a rational s/t is irrational unless s and t are perfect nth powers
negative_passages.docid: 121831
negative_passages.text: Proving that rational numbers are dense
negative_passages.docid: 30411
negative_passages.text: The closure of the irrational numbers on $\mathbb{R}$
negative_passages.docid: 52556
negative_passages.text: Isomorphism of $S^1$.
negative_passages.docid: 30178
negative_passages.text: Homeomorphism $(\mathbb{CP}^1)^m/S_m \overset{\sim}{\to} \mathbb{CP}^m$?
negative_passages.docid: 15695
negative_passages.text: Is ${\rm conv}({\rm ext}((C(X))_1))$ dense in $(C(X))_1$?
negative_passages.docid: 132311
negative_passages.text: T$\mathbb{S}^{n} \times \mathbb{R}$ is diffeomorphic to $\mathbb{S}^{n}\times \mathbb{R}^{n+1}$
negative_passages.docid: 39453
negative_passages.text: How do I prove that $S^n$ is homeomorphic to $S^m \Rightarrow m=n$?
negative_passages.docid: 137030
negative_passages.text: Why are there more Irrationals than Rationals given the density of $Q$ in $R$?
negative_passages.docid: 53945
negative_passages.text: The square root of a prime number is irrational
negative_passages.docid: 15110
negative_passages.text: About irrational logarithms
negative_passages.docid: 35837
negative_passages.text: How to prove that any (integer)$^{1/n}$ that isn't an integer, is irrational?
negative_passages.docid: 108305
negative_passages.text: How can we show that the dyadic rationals are dense in $\mathbb R$?
negative_passages.docid: 35569
negative_passages.text: Prove that $[0,1]$ isn't homeomorphic to $\mathbb{R}$
negative_passages.docid: 33309
negative_passages.text: Subgroup of $\mathbb{R}$ either dense or has a least positive element?
negative_passages.docid: 41617
negative_passages.text: If a $C^1$ map $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is expanding ($\|f(x)-f(y)\|\geq\|x-y\|$), which of these must be true?