Mixing
Clarification on a proof of Roth's theorem
$begingroup$
Roth's theorem is stated in the book by Einsiedler and Ward, theorem 7.14 page 191 as:
Let $(X,mathscr{B},mu,T)$ be a measure-preserving probability system. Then, for any functions $f_1,f_2 in L^{infty}(X,mathscr{B},mu)$, $$frac{1}{N}sumlimits_{n=1}^{N}U_T^{n}f_1U_T^{2n}f_2$$ converges in $L^2(X,mathscr{B},mu)$ (Here $U_T g:=gcirc T$). Moreover, for any $Ain mathscr{B}$ with $mu(A)>0$ we have $$lim_{Nto infty}frac{1}{N}sumlimits_{n=1}^{N} mu(Acap T^{-n}A cap T^{-2n}A)>0.$$
Question: Can I deduce this theorem from the special case where $(X,mathscr{B},mu,T)$ is taken to be an invertible, ergodic, Borel probability system?
The reason I'm asking is that Einsiedler-Ward only seem to prove this for the special case. I'm not sure if i'm misreading their proof or if the reduction to the general case is easy.
My issue is primarily with the $L^2$ convergence claim. A first attempt at the reduction could be to apply the ergodic decomposition theorem. However this isn't a valid approach since our space isn't assumed to be a Borel space.
functional-analysis probability-theory measure-theory ergodic-theory additive-combinatorics
asked Nov 7 '18 at 20:13
Sir Wilfred Lucas-DockerySir Wilfred Lucas-Dockery
414419
$endgroup$
add a comment |
$begingroup$
Roth's theorem is stated in the book by Einsiedler and Ward, theorem 7.14 page 191 as:
Let $(X,mathscr{B},mu,T)$ be a measure-preserving probability system. Then, for any functions $f_1,f_2 in L^{infty}(X,mathscr{B},mu)$, $$frac{1}{N}sumlimits_{n=1}^{N}U_T^{n}f_1U_T^{2n}f_2$$ converges in $L^2(X,mathscr{B},mu)$ (Here $U_T g:=gcirc T$). Moreover, for any $Ain mathscr{B}$ with $mu(A)>0$ we have $$lim_{Nto infty}frac{1}{N}sumlimits_{n=1}^{N} mu(Acap T^{-n}A cap T^{-2n}A)>0.$$
Question: Can I deduce this theorem from the special case where $(X,mathscr{B},mu,T)$ is taken to be an invertible, ergodic, Borel probability system?
The reason I'm asking is that Einsiedler-Ward only seem to prove this for the special case. I'm not sure if i'm misreading their proof or if the reduction to the general case is easy.
My issue is primarily with the $L^2$ convergence claim. A first attempt at the reduction could be to apply the ergodic decomposition theorem. However this isn't a valid approach since our space isn't assumed to be a Borel space.
functional-analysis probability-theory measure-theory ergodic-theory additive-combinatorics
asked Nov 7 '18 at 20:13
Sir Wilfred Lucas-DockerySir Wilfred Lucas-Dockery
414419
$endgroup$
$begingroup$
Just a small remark that doesn't answer the question: you still get Roth's theorem (i.e. large subsets of Z have a 3AP) from the special case.
$endgroup$
– mathworker21
Nov 13 '18 at 2:03
$begingroup$
Yes. Prior to this theorem, the book reduces the proof of Furstenberg's 'multiple recurrence theorem' to the invertible, ergodic, Borel case. But the $L^2$ convergence statement in this theorem seems to be much stronger. I was wondering if a similar reduction could be made.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 13 '18 at 11:24
$begingroup$
what's the issue if you try to go through the reductions that the book makes for Furstenberg's multiple recurrence theorem?
$endgroup$
– mathworker21
Nov 13 '18 at 12:35
1
$begingroup$
one of the steps is to reduce the system to a borel system. For any measurable $A$, you consider a factor of the form ${0,1}^{mathbb{N}}$ with an appropriate measure and factor map. Multiple recurrence follows for the original system if you prove it for all these borel factors. Not sure what the analogous step here would be.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 13 '18 at 15:17
$begingroup$
Moreover, multiple recurrence is a statement involving some limit inferior. If I recall, the reduction to an ergodic system involves using fatous lemma in this lim inf. This theorem Is about convergence. This could be another issue.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 13 '18 at 15:20
add a comment |
$begingroup$
Roth's theorem is stated in the book by Einsiedler and Ward, theorem 7.14 page 191 as:
Let $(X,mathscr{B},mu,T)$ be a measure-preserving probability system. Then, for any functions $f_1,f_2 in L^{infty}(X,mathscr{B},mu)$, $$frac{1}{N}sumlimits_{n=1}^{N}U_T^{n}f_1U_T^{2n}f_2$$ converges in $L^2(X,mathscr{B},mu)$ (Here $U_T g:=gcirc T$). Moreover, for any $Ain mathscr{B}$ with $mu(A)>0$ we have $$lim_{Nto infty}frac{1}{N}sumlimits_{n=1}^{N} mu(Acap T^{-n}A cap T^{-2n}A)>0.$$
Question: Can I deduce this theorem from the special case where $(X,mathscr{B},mu,T)$ is taken to be an invertible, ergodic, Borel probability system?
The reason I'm asking is that Einsiedler-Ward only seem to prove this for the special case. I'm not sure if i'm misreading their proof or if the reduction to the general case is easy.
My issue is primarily with the $L^2$ convergence claim. A first attempt at the reduction could be to apply the ergodic decomposition theorem. However this isn't a valid approach since our space isn't assumed to be a Borel space.
functional-analysis probability-theory measure-theory ergodic-theory additive-combinatorics
asked Nov 7 '18 at 20:13
Sir Wilfred Lucas-DockerySir Wilfred Lucas-Dockery
414419
$endgroup$
Roth's theorem is stated in the book by Einsiedler and Ward, theorem 7.14 page 191 as:
Let $(X,mathscr{B},mu,T)$ be a measure-preserving probability system. Then, for any functions $f_1,f_2 in L^{infty}(X,mathscr{B},mu)$, $$frac{1}{N}sumlimits_{n=1}^{N}U_T^{n}f_1U_T^{2n}f_2$$ converges in $L^2(X,mathscr{B},mu)$ (Here $U_T g:=gcirc T$). Moreover, for any $Ain mathscr{B}$ with $mu(A)>0$ we have $$lim_{Nto infty}frac{1}{N}sumlimits_{n=1}^{N} mu(Acap T^{-n}A cap T^{-2n}A)>0.$$
Question: Can I deduce this theorem from the special case where $(X,mathscr{B},mu,T)$ is taken to be an invertible, ergodic, Borel probability system?
The reason I'm asking is that Einsiedler-Ward only seem to prove this for the special case. I'm not sure if i'm misreading their proof or if the reduction to the general case is easy.
My issue is primarily with the $L^2$ convergence claim. A first attempt at the reduction could be to apply the ergodic decomposition theorem. However this isn't a valid approach since our space isn't assumed to be a Borel space.
functional-analysis probability-theory measure-theory ergodic-theory additive-combinatorics
functional-analysis probability-theory measure-theory ergodic-theory additive-combinatorics
asked Nov 7 '18 at 20:13
Sir Wilfred Lucas-DockerySir Wilfred Lucas-Dockery
414419
asked Nov 7 '18 at 20:13
Sir Wilfred Lucas-DockerySir Wilfred Lucas-Dockery
414419
edited Jan 27 at 21:20
Sir Wilfred Lucas-Dockery
asked Nov 7 '18 at 20:13
Sir Wilfred Lucas-DockerySir Wilfred Lucas-Dockery
414419
asked Nov 7 '18 at 20:13
Sir Wilfred Lucas-DockerySir Wilfred Lucas-Dockery
414419
asked Nov 7 '18 at 20:13
Sir Wilfred Lucas-DockerySir Wilfred Lucas-Dockery
414419
414419
$begingroup$
Just a small remark that doesn't answer the question: you still get Roth's theorem (i.e. large subsets of Z have a 3AP) from the special case.
$endgroup$
– mathworker21
Nov 13 '18 at 2:03
$begingroup$
Yes. Prior to this theorem, the book reduces the proof of Furstenberg's 'multiple recurrence theorem' to the invertible, ergodic, Borel case. But the $L^2$ convergence statement in this theorem seems to be much stronger. I was wondering if a similar reduction could be made.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 13 '18 at 11:24
$begingroup$
what's the issue if you try to go through the reductions that the book makes for Furstenberg's multiple recurrence theorem?
$endgroup$
– mathworker21
Nov 13 '18 at 12:35
1
$begingroup$
one of the steps is to reduce the system to a borel system. For any measurable $A$, you consider a factor of the form ${0,1}^{mathbb{N}}$ with an appropriate measure and factor map. Multiple recurrence follows for the original system if you prove it for all these borel factors. Not sure what the analogous step here would be.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 13 '18 at 15:17
$begingroup$
Moreover, multiple recurrence is a statement involving some limit inferior. If I recall, the reduction to an ergodic system involves using fatous lemma in this lim inf. This theorem Is about convergence. This could be another issue.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 13 '18 at 15:20
add a comment |
$begingroup$
Just a small remark that doesn't answer the question: you still get Roth's theorem (i.e. large subsets of Z have a 3AP) from the special case.
$endgroup$
– mathworker21
Nov 13 '18 at 2:03
$begingroup$
Yes. Prior to this theorem, the book reduces the proof of Furstenberg's 'multiple recurrence theorem' to the invertible, ergodic, Borel case. But the $L^2$ convergence statement in this theorem seems to be much stronger. I was wondering if a similar reduction could be made.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 13 '18 at 11:24
$begingroup$
what's the issue if you try to go through the reductions that the book makes for Furstenberg's multiple recurrence theorem?
$endgroup$
– mathworker21
Nov 13 '18 at 12:35
1
$begingroup$
one of the steps is to reduce the system to a borel system. For any measurable $A$, you consider a factor of the form ${0,1}^{mathbb{N}}$ with an appropriate measure and factor map. Multiple recurrence follows for the original system if you prove it for all these borel factors. Not sure what the analogous step here would be.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 13 '18 at 15:17
$begingroup$
Moreover, multiple recurrence is a statement involving some limit inferior. If I recall, the reduction to an ergodic system involves using fatous lemma in this lim inf. This theorem Is about convergence. This could be another issue.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 13 '18 at 15:20
$begingroup$
Just a small remark that doesn't answer the question: you still get Roth's theorem (i.e. large subsets of Z have a 3AP) from the special case.
$endgroup$
– mathworker21
Nov 13 '18 at 2:03
$begingroup$
Just a small remark that doesn't answer the question: you still get Roth's theorem (i.e. large subsets of Z have a 3AP) from the special case.
$endgroup$
– mathworker21
Nov 13 '18 at 2:03
$begingroup$
Yes. Prior to this theorem, the book reduces the proof of Furstenberg's 'multiple recurrence theorem' to the invertible, ergodic, Borel case. But the $L^2$ convergence statement in this theorem seems to be much stronger. I was wondering if a similar reduction could be made.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 13 '18 at 11:24
$begingroup$
Yes. Prior to this theorem, the book reduces the proof of Furstenberg's 'multiple recurrence theorem' to the invertible, ergodic, Borel case. But the $L^2$ convergence statement in this theorem seems to be much stronger. I was wondering if a similar reduction could be made.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 13 '18 at 11:24
$begingroup$
what's the issue if you try to go through the reductions that the book makes for Furstenberg's multiple recurrence theorem?
$endgroup$
– mathworker21
Nov 13 '18 at 12:35
$begingroup$
what's the issue if you try to go through the reductions that the book makes for Furstenberg's multiple recurrence theorem?
$endgroup$
– mathworker21
Nov 13 '18 at 12:35
1
1
$begingroup$
one of the steps is to reduce the system to a borel system. For any measurable $A$, you consider a factor of the form ${0,1}^{mathbb{N}}$ with an appropriate measure and factor map. Multiple recurrence follows for the original system if you prove it for all these borel factors. Not sure what the analogous step here would be.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 13 '18 at 15:17
$begingroup$
one of the steps is to reduce the system to a borel system. For any measurable $A$, you consider a factor of the form ${0,1}^{mathbb{N}}$ with an appropriate measure and factor map. Multiple recurrence follows for the original system if you prove it for all these borel factors. Not sure what the analogous step here would be.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 13 '18 at 15:17
$begingroup$
Moreover, multiple recurrence is a statement involving some limit inferior. If I recall, the reduction to an ergodic system involves using fatous lemma in this lim inf. This theorem Is about convergence. This could be another issue.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 13 '18 at 15:20
$begingroup$
Moreover, multiple recurrence is a statement involving some limit inferior. If I recall, the reduction to an ergodic system involves using fatous lemma in this lim inf. This theorem Is about convergence. This could be another issue.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 13 '18 at 15:20
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%2f2989042%2fclarification-on-a-proof-of-roths-theorem%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%2f2989042%2fclarification-on-a-proof-of-roths-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
$begingroup$
Just a small remark that doesn't answer the question: you still get Roth's theorem (i.e. large subsets of Z have a 3AP) from the special case.
$endgroup$
– mathworker21
Nov 13 '18 at 2:03
$begingroup$
Yes. Prior to this theorem, the book reduces the proof of Furstenberg's 'multiple recurrence theorem' to the invertible, ergodic, Borel case. But the $L^2$ convergence statement in this theorem seems to be much stronger. I was wondering if a similar reduction could be made.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 13 '18 at 11:24
$begingroup$
what's the issue if you try to go through the reductions that the book makes for Furstenberg's multiple recurrence theorem?
$endgroup$
– mathworker21
Nov 13 '18 at 12:35
1
$begingroup$
one of the steps is to reduce the system to a borel system. For any measurable $A$, you consider a factor of the form ${0,1}^{mathbb{N}}$ with an appropriate measure and factor map. Multiple recurrence follows for the original system if you prove it for all these borel factors. Not sure what the analogous step here would be.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 13 '18 at 15:17
$begingroup$
Moreover, multiple recurrence is a statement involving some limit inferior. If I recall, the reduction to an ergodic system involves using fatous lemma in this lim inf. This theorem Is about convergence. This could be another issue.
$endgroup$
– Sir Wilfred Lucas-Dockery
Nov 13 '18 at 15:20