Question

For each $n\in\mathbb{N},$ let $x_n:=\min_{1\leq k < n}\lvert\sin n-\sin k\rvert.\ $ Does $\sum_{n=1}^{\infty} x_n $ converge?

score 312 · Answer 1 · 2026-03-25 14:25:30

312

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For each $n\in\mathbb{N},$ let $x_n:=\min_{1\leq k < n}\lvert\sin n-\sin k\rvert.\ $ Does $\sum_{n=1}^{\infty} x_n $ converge?

Published on 25 Mar 2026 - 14:25

score 47 · Answer 2 · 2024-02-13 18:11:31

47

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Tighten Corollary of Dirichlet's Simultaneous Approximation Bound (d = 3)

Published on 13 Feb 2024 - 18:11

score 61 · Answer 3 · 2024-02-29 00:10:01

61

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Examples of specific points $g\in G:=\text{SL}(2,\mathbb R)$ such that the curve $u_tg \Gamma, \Gamma:=\text{SL}(2,\mathbb Z)$ is dense in $G/\Gamma$?

Published on 29 Feb 2024 - 0:10

score 83 · Answer 4 · 2024-03-31 18:54:05

If $x,y\in\mathbb{N},\varepsilon>0$ then are there infinitely many positive integer pairs $(n,m)$ s.t. $\vert\frac{x^n}{y^m}- 1\vert < \varepsilon?$

score 2.1k · Answer 5 · 2015-01-05 19:57:34

2.1k

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How was this approximation of $\pi$ involving $\sqrt{5}$ arrived at?

Published on 05 Jan 2015 - 19:57

score 678 · Answer 6 · 2015-02-05 15:53:31

678

Views

Closest rational approximation of $\sqrt x$ with denominator having prime powers $\lt n$

Published on 05 Feb 2015 - 15:53

score 382 · Answer 7 · 2015-02-17 01:27:16

382

Views

Thue's Theorem of Diophantine approximation

Published on 17 Feb 2015 - 1:27

score 69 · Answer 8 · 2015-02-20 02:28:15

69

Views

Guessing a numerical middle bound

Published on 20 Feb 2015 - 2:28

score 589 · Answer 9 · 2015-03-09 17:07:50

589

Views

Lower bound on $|a+b \sqrt{2} + c \sqrt{3}|$

Published on 09 Mar 2015 - 17:07

score 471 · Answer 10 · 2015-03-26 06:22:52

471

Views

Liouville's and Roth's theorems for complex algebraic numbers

Published on 26 Mar 2015 - 6:22

score 420 · Answer 11 · 2015-03-31 19:42:22

420

Views

On a theorem of Kronecker!

Published on 31 Mar 2015 - 19:42

score 2.9k · Answer 12 · 2015-04-09 13:07:54

2.9k

Views

Rational number that approximates $\sqrt{3}$

Published on 09 Apr 2015 - 13:07

score 60 · Answer 13 · 2015-04-17 19:08:51

60

Views

Does there exist a quadratic generalization of the continued fraction approximants?

Published on 17 Apr 2015 - 19:08

score 111 · Answer 14 · 2015-04-21 11:44:13

111

Views

Closeness of $n! \ x$ to integers for irrational $x$

Published on 21 Apr 2015 - 11:44

score 83 · Answer 15 · 2015-04-21 17:52:29

83

Views

How to write $\sqrt{4x^2 - 3}$ in the ring $\mathbb{Q}[x]/(x^3 - x - 1)$?

Published on 21 Apr 2015 - 17:52

List Question

For each $n\in\mathbb{N},$ let $x_n:=\min_{1\leq k < n}\lvert\sin n-\sin k\rvert.\ $ Does $\sum_{n=1}^{\infty} x_n $ converge?

Tighten Corollary of Dirichlet's Simultaneous Approximation Bound (d = 3)

Examples of specific points $g\in G:=\text{SL}(2,\mathbb R)$ such that the curve $u_tg \Gamma, \Gamma:=\text{SL}(2,\mathbb Z)$ is dense in $G/\Gamma$?

If $x,y\in\mathbb{N},\varepsilon>0$ then are there infinitely many positive integer pairs $(n,m)$ s.t. $\vert\frac{x^n}{y^m}- 1\vert < \varepsilon?$

How was this approximation of $\pi$ involving $\sqrt{5}$ arrived at?

Closest rational approximation of $\sqrt x$ with denominator having prime powers $\lt n$

Thue's Theorem of Diophantine approximation

Guessing a numerical middle bound

Lower bound on $|a+b \sqrt{2} + c \sqrt{3}|$

Liouville's and Roth's theorems for complex algebraic numbers

On a theorem of Kronecker!

Rational number that approximates $\sqrt{3}$

Does there exist a quadratic generalization of the continued fraction approximants?

Closeness of $n! \ x$ to integers for irrational $x$

How to write $\sqrt{4x^2 - 3}$ in the ring $\mathbb{Q}[x]/(x^3 - x - 1)$?

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