Special cases of Szemeredi's Theorem?
Are there examples of special cases of Szemeredi's theorem which one can give, which are slightly non-trivial?
To clarify, I'm looking for sets of integers where we can show that they contain infinitely many arithmetic progressions of any finite length. When I say "slightly non-trivial", this is a non-well-defined condition which attempts to remove commenters saying things like "the even numbers" or "all infinite arithmetic progressions" - I'm looking for something like a proof for the special case of the square-free integers.
(Final comment: This is less of a question that I particularly need answered but more of an attempt to collate a collection of such "slightly non-trivial" proofs for my and others' appreciation.)
combinatorics number-theory soft-question alternative-proof arithmetic-progressions
add a comment |
Are there examples of special cases of Szemeredi's theorem which one can give, which are slightly non-trivial?
To clarify, I'm looking for sets of integers where we can show that they contain infinitely many arithmetic progressions of any finite length. When I say "slightly non-trivial", this is a non-well-defined condition which attempts to remove commenters saying things like "the even numbers" or "all infinite arithmetic progressions" - I'm looking for something like a proof for the special case of the square-free integers.
(Final comment: This is less of a question that I particularly need answered but more of an attempt to collate a collection of such "slightly non-trivial" proofs for my and others' appreciation.)
combinatorics number-theory soft-question alternative-proof arithmetic-progressions
add a comment |
Are there examples of special cases of Szemeredi's theorem which one can give, which are slightly non-trivial?
To clarify, I'm looking for sets of integers where we can show that they contain infinitely many arithmetic progressions of any finite length. When I say "slightly non-trivial", this is a non-well-defined condition which attempts to remove commenters saying things like "the even numbers" or "all infinite arithmetic progressions" - I'm looking for something like a proof for the special case of the square-free integers.
(Final comment: This is less of a question that I particularly need answered but more of an attempt to collate a collection of such "slightly non-trivial" proofs for my and others' appreciation.)
combinatorics number-theory soft-question alternative-proof arithmetic-progressions
Are there examples of special cases of Szemeredi's theorem which one can give, which are slightly non-trivial?
To clarify, I'm looking for sets of integers where we can show that they contain infinitely many arithmetic progressions of any finite length. When I say "slightly non-trivial", this is a non-well-defined condition which attempts to remove commenters saying things like "the even numbers" or "all infinite arithmetic progressions" - I'm looking for something like a proof for the special case of the square-free integers.
(Final comment: This is less of a question that I particularly need answered but more of an attempt to collate a collection of such "slightly non-trivial" proofs for my and others' appreciation.)
combinatorics number-theory soft-question alternative-proof arithmetic-progressions
combinatorics number-theory soft-question alternative-proof arithmetic-progressions
asked yesterday
Isky Mathews
878214
878214
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
The Green-Tao theorem comes to mind. It states that the primes contain arithmetic progressions of arbitrary length. If you want to search for other 'non-trivial' special cases, starting with this Erdos conjecture might be a good idea.
There are also really interesting special cases of Szemeredi's Theorem that were proved before and are more accessible such as Roth's Theorem. This theorem states that any subset of integers of positive density contains a $3$ term arithmetic progression. This theorem is particularly interesting because there has been work on improving the density value. A good survey article is here.
While I thank you for the time taken to write your answer, what I meant by slightly non-trivial is that the proofs in question are to be short enough to be contained in a MSE post, assuming potentially certain standard NT theorems (like PNT or Stirling's Approximation). Green-Tao's proof is extremely complex and furthermore assumes Szemeredi's Theorem as part of the proof, so definitely isn't a special case.
– Isky Mathews
yesterday
add a comment |
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%2f3060652%2fspecial-cases-of-szemeredis-theorem%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
The Green-Tao theorem comes to mind. It states that the primes contain arithmetic progressions of arbitrary length. If you want to search for other 'non-trivial' special cases, starting with this Erdos conjecture might be a good idea.
There are also really interesting special cases of Szemeredi's Theorem that were proved before and are more accessible such as Roth's Theorem. This theorem states that any subset of integers of positive density contains a $3$ term arithmetic progression. This theorem is particularly interesting because there has been work on improving the density value. A good survey article is here.
While I thank you for the time taken to write your answer, what I meant by slightly non-trivial is that the proofs in question are to be short enough to be contained in a MSE post, assuming potentially certain standard NT theorems (like PNT or Stirling's Approximation). Green-Tao's proof is extremely complex and furthermore assumes Szemeredi's Theorem as part of the proof, so definitely isn't a special case.
– Isky Mathews
yesterday
add a comment |
The Green-Tao theorem comes to mind. It states that the primes contain arithmetic progressions of arbitrary length. If you want to search for other 'non-trivial' special cases, starting with this Erdos conjecture might be a good idea.
There are also really interesting special cases of Szemeredi's Theorem that were proved before and are more accessible such as Roth's Theorem. This theorem states that any subset of integers of positive density contains a $3$ term arithmetic progression. This theorem is particularly interesting because there has been work on improving the density value. A good survey article is here.
While I thank you for the time taken to write your answer, what I meant by slightly non-trivial is that the proofs in question are to be short enough to be contained in a MSE post, assuming potentially certain standard NT theorems (like PNT or Stirling's Approximation). Green-Tao's proof is extremely complex and furthermore assumes Szemeredi's Theorem as part of the proof, so definitely isn't a special case.
– Isky Mathews
yesterday
add a comment |
The Green-Tao theorem comes to mind. It states that the primes contain arithmetic progressions of arbitrary length. If you want to search for other 'non-trivial' special cases, starting with this Erdos conjecture might be a good idea.
There are also really interesting special cases of Szemeredi's Theorem that were proved before and are more accessible such as Roth's Theorem. This theorem states that any subset of integers of positive density contains a $3$ term arithmetic progression. This theorem is particularly interesting because there has been work on improving the density value. A good survey article is here.
The Green-Tao theorem comes to mind. It states that the primes contain arithmetic progressions of arbitrary length. If you want to search for other 'non-trivial' special cases, starting with this Erdos conjecture might be a good idea.
There are also really interesting special cases of Szemeredi's Theorem that were proved before and are more accessible such as Roth's Theorem. This theorem states that any subset of integers of positive density contains a $3$ term arithmetic progression. This theorem is particularly interesting because there has been work on improving the density value. A good survey article is here.
answered yesterday
Sandeep Silwal
5,84311236
5,84311236
While I thank you for the time taken to write your answer, what I meant by slightly non-trivial is that the proofs in question are to be short enough to be contained in a MSE post, assuming potentially certain standard NT theorems (like PNT or Stirling's Approximation). Green-Tao's proof is extremely complex and furthermore assumes Szemeredi's Theorem as part of the proof, so definitely isn't a special case.
– Isky Mathews
yesterday
add a comment |
While I thank you for the time taken to write your answer, what I meant by slightly non-trivial is that the proofs in question are to be short enough to be contained in a MSE post, assuming potentially certain standard NT theorems (like PNT or Stirling's Approximation). Green-Tao's proof is extremely complex and furthermore assumes Szemeredi's Theorem as part of the proof, so definitely isn't a special case.
– Isky Mathews
yesterday
While I thank you for the time taken to write your answer, what I meant by slightly non-trivial is that the proofs in question are to be short enough to be contained in a MSE post, assuming potentially certain standard NT theorems (like PNT or Stirling's Approximation). Green-Tao's proof is extremely complex and furthermore assumes Szemeredi's Theorem as part of the proof, so definitely isn't a special case.
– Isky Mathews
yesterday
While I thank you for the time taken to write your answer, what I meant by slightly non-trivial is that the proofs in question are to be short enough to be contained in a MSE post, assuming potentially certain standard NT theorems (like PNT or Stirling's Approximation). Green-Tao's proof is extremely complex and furthermore assumes Szemeredi's Theorem as part of the proof, so definitely isn't a special case.
– Isky Mathews
yesterday
add a comment |
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3060652%2fspecial-cases-of-szemeredis-theorem%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