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
$endgroup$
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
$endgroup$
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
$endgroup$
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
edited Dec 29 '18 at 11:06
Yoav Sternberg
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 |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3055270%2fis-the-set-n-cosn-n-sinn-dense-in-0-12%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3055270%2fis-the-set-n-cosn-n-sinn-dense-in-0-12%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
$begingroup$
Related: Is $nsin n$ dense on the real line?
$endgroup$
– Winther
Dec 28 '18 at 20:56