Hölder criterion and Hausdorff dimension
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I've recently learned that the Cantor function is Hölder continuous with parameter $alpha$ where $alpha$ is the Hausdorff dimension of the Cantor set.
My intuition is that this fact stems from the fact that the Cantor function is essentially the "CDF corresponding to random choice from the Cantor set", I've been wondering if this intuition can be generalized as follows:
Let there be $0<D<1$ and let $Csubsetmathbb{R}$ be a set of Hausdorff dimension $D$ such that $mathcal{H}_{D}(C)=1$.
Define a function: $$f(x)=mathcal{H}_{D}((-infty,x)cap C)$$
is $f$ necessarily Hölder continuous with parameter $D$?
real-analysis hausdorff-measure
asked Dec 7 '18 at 18:29
Bar AlonBar Alon
484114
$endgroup$
add a comment |
$begingroup$
I've recently learned that the Cantor function is Hölder continuous with parameter $alpha$ where $alpha$ is the Hausdorff dimension of the Cantor set.
My intuition is that this fact stems from the fact that the Cantor function is essentially the "CDF corresponding to random choice from the Cantor set", I've been wondering if this intuition can be generalized as follows:
Let there be $0<D<1$ and let $Csubsetmathbb{R}$ be a set of Hausdorff dimension $D$ such that $mathcal{H}_{D}(C)=1$.
Define a function: $$f(x)=mathcal{H}_{D}((-infty,x)cap C)$$
is $f$ necessarily Hölder continuous with parameter $D$?
real-analysis hausdorff-measure
asked Dec 7 '18 at 18:29
Bar AlonBar Alon
484114
$endgroup$
2
$begingroup$
Just a random thought, without much thought to your question (I'm skimming over questions quickly during a spare moment), but it seems to me that one might make this fail by scrunching up most of the $D$-dimensional measure of $C$ in one location, and if so, then a natural follow-up question is whether Hölder continuity follows if $C cap (a,b)$ has $D$-measure equal to $b-a$ for every $0 < a < b leq text{diam}(C).$
$endgroup$
– Dave L. Renfro
Dec 7 '18 at 19:30
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@DaveL.Renfro I've had a thought along those lines too, but I couldn't come up with a simple way of doing this without decreasing the Hausdorff dimension
$endgroup$
– Bar Alon
Dec 9 '18 at 11:01
1
$begingroup$
I happened to come across this paper today (by accident, while looking for something else) --- On distribution of local dimensions of doubling measures on Euclidean space by Peng/Wen (2017) --- and the abstract suggests that it might be slightly relevant to what you're asking. The paper is behind a paywall (for me, at least), and there doesn't appear to be a preprint version posted anywhere, although I haven't looked very hard for one.
$endgroup$
– Dave L. Renfro
Dec 23 '18 at 21:33
add a comment |
$begingroup$
I've recently learned that the Cantor function is Hölder continuous with parameter $alpha$ where $alpha$ is the Hausdorff dimension of the Cantor set.
My intuition is that this fact stems from the fact that the Cantor function is essentially the "CDF corresponding to random choice from the Cantor set", I've been wondering if this intuition can be generalized as follows:
Let there be $0<D<1$ and let $Csubsetmathbb{R}$ be a set of Hausdorff dimension $D$ such that $mathcal{H}_{D}(C)=1$.
Define a function: $$f(x)=mathcal{H}_{D}((-infty,x)cap C)$$
is $f$ necessarily Hölder continuous with parameter $D$?
real-analysis hausdorff-measure
asked Dec 7 '18 at 18:29
Bar AlonBar Alon
484114
$endgroup$
I've recently learned that the Cantor function is Hölder continuous with parameter $alpha$ where $alpha$ is the Hausdorff dimension of the Cantor set.
My intuition is that this fact stems from the fact that the Cantor function is essentially the "CDF corresponding to random choice from the Cantor set", I've been wondering if this intuition can be generalized as follows:
Let there be $0<D<1$ and let $Csubsetmathbb{R}$ be a set of Hausdorff dimension $D$ such that $mathcal{H}_{D}(C)=1$.
Define a function: $$f(x)=mathcal{H}_{D}((-infty,x)cap C)$$
is $f$ necessarily Hölder continuous with parameter $D$?
real-analysis hausdorff-measure
real-analysis hausdorff-measure
asked Dec 7 '18 at 18:29
Bar AlonBar Alon
484114
asked Dec 7 '18 at 18:29
Bar AlonBar Alon
484114
asked Dec 7 '18 at 18:29
Bar AlonBar Alon
484114
asked Dec 7 '18 at 18:29
Bar AlonBar Alon
484114
asked Dec 7 '18 at 18:29
Bar AlonBar Alon
484114
484114
2
$begingroup$
Just a random thought, without much thought to your question (I'm skimming over questions quickly during a spare moment), but it seems to me that one might make this fail by scrunching up most of the $D$-dimensional measure of $C$ in one location, and if so, then a natural follow-up question is whether Hölder continuity follows if $C cap (a,b)$ has $D$-measure equal to $b-a$ for every $0 < a < b leq text{diam}(C).$
$endgroup$
– Dave L. Renfro
Dec 7 '18 at 19:30
$begingroup$
@DaveL.Renfro I've had a thought along those lines too, but I couldn't come up with a simple way of doing this without decreasing the Hausdorff dimension
$endgroup$
– Bar Alon
Dec 9 '18 at 11:01
1
$begingroup$
I happened to come across this paper today (by accident, while looking for something else) --- On distribution of local dimensions of doubling measures on Euclidean space by Peng/Wen (2017) --- and the abstract suggests that it might be slightly relevant to what you're asking. The paper is behind a paywall (for me, at least), and there doesn't appear to be a preprint version posted anywhere, although I haven't looked very hard for one.
$endgroup$
– Dave L. Renfro
Dec 23 '18 at 21:33
add a comment |
2
$begingroup$
Just a random thought, without much thought to your question (I'm skimming over questions quickly during a spare moment), but it seems to me that one might make this fail by scrunching up most of the $D$-dimensional measure of $C$ in one location, and if so, then a natural follow-up question is whether Hölder continuity follows if $C cap (a,b)$ has $D$-measure equal to $b-a$ for every $0 < a < b leq text{diam}(C).$
$endgroup$
– Dave L. Renfro
Dec 7 '18 at 19:30
$begingroup$
@DaveL.Renfro I've had a thought along those lines too, but I couldn't come up with a simple way of doing this without decreasing the Hausdorff dimension
$endgroup$
– Bar Alon
Dec 9 '18 at 11:01
1
$begingroup$
I happened to come across this paper today (by accident, while looking for something else) --- On distribution of local dimensions of doubling measures on Euclidean space by Peng/Wen (2017) --- and the abstract suggests that it might be slightly relevant to what you're asking. The paper is behind a paywall (for me, at least), and there doesn't appear to be a preprint version posted anywhere, although I haven't looked very hard for one.
$endgroup$
– Dave L. Renfro
Dec 23 '18 at 21:33
2
2
$begingroup$
Just a random thought, without much thought to your question (I'm skimming over questions quickly during a spare moment), but it seems to me that one might make this fail by scrunching up most of the $D$-dimensional measure of $C$ in one location, and if so, then a natural follow-up question is whether Hölder continuity follows if $C cap (a,b)$ has $D$-measure equal to $b-a$ for every $0 < a < b leq text{diam}(C).$
$endgroup$
– Dave L. Renfro
Dec 7 '18 at 19:30
$begingroup$
Just a random thought, without much thought to your question (I'm skimming over questions quickly during a spare moment), but it seems to me that one might make this fail by scrunching up most of the $D$-dimensional measure of $C$ in one location, and if so, then a natural follow-up question is whether Hölder continuity follows if $C cap (a,b)$ has $D$-measure equal to $b-a$ for every $0 < a < b leq text{diam}(C).$
$endgroup$
– Dave L. Renfro
Dec 7 '18 at 19:30
$begingroup$
@DaveL.Renfro I've had a thought along those lines too, but I couldn't come up with a simple way of doing this without decreasing the Hausdorff dimension
$endgroup$
– Bar Alon
Dec 9 '18 at 11:01
$begingroup$
@DaveL.Renfro I've had a thought along those lines too, but I couldn't come up with a simple way of doing this without decreasing the Hausdorff dimension
$endgroup$
– Bar Alon
Dec 9 '18 at 11:01
1
1
$begingroup$
I happened to come across this paper today (by accident, while looking for something else) --- On distribution of local dimensions of doubling measures on Euclidean space by Peng/Wen (2017) --- and the abstract suggests that it might be slightly relevant to what you're asking. The paper is behind a paywall (for me, at least), and there doesn't appear to be a preprint version posted anywhere, although I haven't looked very hard for one.
$endgroup$
– Dave L. Renfro
Dec 23 '18 at 21:33
$begingroup$
I happened to come across this paper today (by accident, while looking for something else) --- On distribution of local dimensions of doubling measures on Euclidean space by Peng/Wen (2017) --- and the abstract suggests that it might be slightly relevant to what you're asking. The paper is behind a paywall (for me, at least), and there doesn't appear to be a preprint version posted anywhere, although I haven't looked very hard for one.
$endgroup$
– Dave L. Renfro
Dec 23 '18 at 21:33
add a comment |
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$begingroup$
Just a random thought, without much thought to your question (I'm skimming over questions quickly during a spare moment), but it seems to me that one might make this fail by scrunching up most of the $D$-dimensional measure of $C$ in one location, and if so, then a natural follow-up question is whether Hölder continuity follows if $C cap (a,b)$ has $D$-measure equal to $b-a$ for every $0 < a < b leq text{diam}(C).$
$endgroup$
– Dave L. Renfro
Dec 7 '18 at 19:30
$begingroup$
@DaveL.Renfro I've had a thought along those lines too, but I couldn't come up with a simple way of doing this without decreasing the Hausdorff dimension
$endgroup$
– Bar Alon
Dec 9 '18 at 11:01
1
$begingroup$
I happened to come across this paper today (by accident, while looking for something else) --- On distribution of local dimensions of doubling measures on Euclidean space by Peng/Wen (2017) --- and the abstract suggests that it might be slightly relevant to what you're asking. The paper is behind a paywall (for me, at least), and there doesn't appear to be a preprint version posted anywhere, although I haven't looked very hard for one.
$endgroup$
– Dave L. Renfro
Dec 23 '18 at 21:33