query: For $x\in\mathbb R\setminus\mathbb Q$, the set $\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$ is dense on $[0,1)$
positive_passages.docid: 81150
positive_passages.text: Positive integer multiples of an irrational mod 1 are dense
negative_passages.docid: 146023
negative_passages.text: How to prove that $x^2 - \lfloor x\rfloor^2$ is onto?
negative_passages.docid: 15695
negative_passages.text: Is ${\rm conv}({\rm ext}((C(X))_1))$ dense in $(C(X))_1$?
negative_passages.docid: 77129
negative_passages.text: Density of irrationals
negative_passages.docid: 31990
negative_passages.text: Is $\{\frac{m}{10^n}\mid m,n\in\mathbb Z,\ n\geq 0\}$ dense in $\mathbb R$?
negative_passages.docid: 78915
negative_passages.text: Density of the set $S=\{m/2^n| n\in\mathbb{N}, m\in\mathbb{Z}\}$ on $\mathbb{R}$?
negative_passages.docid: 104234
negative_passages.text: What is the Hausdorff dimension of the set of irrationals on $\mathbb R^1$
negative_passages.docid: 116542
negative_passages.text: Which of the following sets are compact in $\mathbb{M}_n(\mathbb{R})$[NBHM_PhD Screening Test-2013, Topology]
negative_passages.docid: 77973
negative_passages.text: How to prove or disprove $\forall x\in\Bbb{R}, \forall n\in\Bbb{N},n\gt 0\implies \lfloor\frac{\lfloor x\rfloor}{n}\rfloor=\lfloor\frac{x}{n}\rfloor$.
negative_passages.docid: 14048
negative_passages.text: Proof that the complement of zero-measure set on $[0,1]$ is dense in $[0,1]$
negative_passages.docid: 35105
negative_passages.text: Density of ring of algebraic integers in $\mathbb C$
negative_passages.docid: 135814
negative_passages.text: Prove certain subsets of $\mathbb R^n$ are subspaces
negative_passages.docid: 137081
negative_passages.text: Is the set of real matrices diagonalizable in $\mathcal M_n(C)$ dense in the set of all companion matrices?
negative_passages.docid: 12212
negative_passages.text: Showing a certain subspace of Hilbert space is dense
negative_passages.docid: 130297
negative_passages.text: $A$ be a subset of $[0,1]$ with non-empty interior ; then is it true that $\mathbb Q+A=\mathbb R$?
negative_passages.docid: 102250
negative_passages.text: Subset of $C[0,1]$ is nowhere dense
negative_passages.docid: 158582
negative_passages.text: Prove that $\{\frac{\phi (n)}{n}\}_{n \in \Bbb N}$ is dense in $[0,1]$