Is the set $({ncos(n)},{nsin(n)})$ dense in $[0,1]^2$
$begingroup$
Let ${cdot}$ be the fractional part. $Bbb N$ is the set of positive integers.
Is the set ${ ({n cos(n)}, {n sin(n)}): n in mathbb{N}}$ dense in $[0,1]^2$?
It is known that the set ${ {sin{n}} : n in mathbb{N}}$ is dense in $[0,1]$, and the same about $cos(n)$. In addition, according to this answer, ${n sin(n): n in mathbb{N}}$ is dense in $mathbb{R}$ under a
"reasonable" assumption.
I've also plotted the first 1,000,000 points to see if it "seems to be dense".
real-analysis real-numbers
asked Dec 28 '18 at 20:48
Yoav SternbergYoav Sternberg
1044
$endgroup$
add a comment |
$begingroup$
Let ${cdot}$ be the fractional part. $Bbb N$ is the set of positive integers.
Is the set ${ ({n cos(n)}, {n sin(n)}): n in mathbb{N}}$ dense in $[0,1]^2$?
It is known that the set ${ {sin{n}} : n in mathbb{N}}$ is dense in $[0,1]$, and the same about $cos(n)$. In addition, according to this answer, ${n sin(n): n in mathbb{N}}$ is dense in $mathbb{R}$ under a
"reasonable" assumption.
I've also plotted the first 1,000,000 points to see if it "seems to be dense".
real-analysis real-numbers
asked Dec 28 '18 at 20:48
Yoav SternbergYoav Sternberg
1044
$endgroup$
1
$begingroup$
Related: Is $nsin n$ dense on the real line?
$endgroup$
– Winther
Dec 28 '18 at 20:56
add a comment |
$begingroup$
Let ${cdot}$ be the fractional part. $Bbb N$ is the set of positive integers.
Is the set ${ ({n cos(n)}, {n sin(n)}): n in mathbb{N}}$ dense in $[0,1]^2$?
It is known that the set ${ {sin{n}} : n in mathbb{N}}$ is dense in $[0,1]$, and the same about $cos(n)$. In addition, according to this answer, ${n sin(n): n in mathbb{N}}$ is dense in $mathbb{R}$ under a
"reasonable" assumption.
I've also plotted the first 1,000,000 points to see if it "seems to be dense".
real-analysis real-numbers
asked Dec 28 '18 at 20:48
Yoav SternbergYoav Sternberg
1044
$endgroup$
Let ${cdot}$ be the fractional part. $Bbb N$ is the set of positive integers.
Is the set ${ ({n cos(n)}, {n sin(n)}): n in mathbb{N}}$ dense in $[0,1]^2$?
It is known that the set ${ {sin{n}} : n in mathbb{N}}$ is dense in $[0,1]$, and the same about $cos(n)$. In addition, according to this answer, ${n sin(n): n in mathbb{N}}$ is dense in $mathbb{R}$ under a
"reasonable" assumption.
I've also plotted the first 1,000,000 points to see if it "seems to be dense".
real-analysis real-numbers
real-analysis real-numbers
asked Dec 28 '18 at 20:48
Yoav SternbergYoav Sternberg
1044
asked Dec 28 '18 at 20:48
Yoav SternbergYoav Sternberg
1044
edited Dec 29 '18 at 11:06
Yoav Sternberg
asked Dec 28 '18 at 20:48
Yoav SternbergYoav Sternberg
1044
asked Dec 28 '18 at 20:48
Yoav SternbergYoav Sternberg
1044
asked Dec 28 '18 at 20:48
Yoav SternbergYoav Sternberg
1044
1044
1
$begingroup$
Related: Is $nsin n$ dense on the real line?
$endgroup$
– Winther
Dec 28 '18 at 20:56
add a comment |
1
$begingroup$
Related: Is $nsin n$ dense on the real line?
$endgroup$
– Winther
Dec 28 '18 at 20:56
1
1
$begingroup$
Related: Is $nsin n$ dense on the real line?
$endgroup$
– Winther
Dec 28 '18 at 20:56
$begingroup$
Related: Is $nsin n$ dense on the real line?
$endgroup$
– Winther
Dec 28 '18 at 20:56
add a comment |
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Related: Is $nsin n$ dense on the real line?
$endgroup$
– Winther
Dec 28 '18 at 20:56