# Hausdorff Dimension of Set of Measure Zero

It's clear that every $A \subset \mathbb R^n$ with $\dim_H(A) < n$ we have $\mathcal H^n(A) = 0$. Is there any $A \subset \mathbb R^n$ with $\mathcal H^n(A) = 0$ but $\dim_H(A) = n$? Thank you.

• $\mathcal H^n$ means the Lebesgue measure, doesn't it? Sep 20, 2012 at 14:02
• $n$-dimensional Hausdorff measure. Yes it's equal to $n$-dimensional Lebesgue measure (with some scaling).
– Deco
Sep 20, 2012 at 14:06
• The Hausdorff measure $H^d(X)$ with $d=\dim_H(X)$ can be finite or zero or infinite. (As far as I know) Sep 20, 2012 at 14:34
• Intuitively, it's hard to imagine a set of measure zero has "full" dimension. But, a proof is still needed.
– Deco
Sep 21, 2012 at 1:46
• See a previous answer... math.stackexchange.com/a/132485/442 Sep 22, 2012 at 2:18

You can get a concrete example of such a set for the case $n=1$ by making a slight modification to the construction of the ternary Cantor set. Instead of taking away $\frac13$ of every interval at each step, remove $\frac{1}{k+1}$ in step $k$.

We can calculate the $d$-dimensional Hausdorff measure as follows. After $k$ steps we are left with $2^k$ intervals with a total length of $$\frac12 \cdot \frac23 \cdot \frac34 \cdots \frac{k}{k+1} = \frac{1}{k+1}$$ With some hand-waving we can then find the measure as $$\mathcal{H}^d(A) = \lim_{k\to\infty} 2^k \left(\frac{1}{(k+1)2^k}\right)^d = \lim_{k\to\infty} 2^{(1-d)k}(k+1)^{-d}$$ It is clear that for $1-d > 0$ the exponential beats the power function, making the measure infinite, but at $d=1$ the limit is $0$.

Note that this doesn't mean you can't get a nontrivial Hausdorff measure on this set. You just need to use a test function other than $x^d$. In this case I think $-x \log_2 x$ will work.
The text passage discusses the Hausdorff dimension and measure of subsets of $\mathbb{R}^n$, specifically asking if there can be a set with Hausdorff dimension equal to $n$ but with $n$-dimensional Hausdorff measure zero. It also provides an example of such a set for $n=1$ by modifying the construction of the Cantor set.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Hausdorff Dimension of Set of Measure Zero

It's clear that every $A \subset \mathbb R^n$ with $\dim_H(A) < n$ we have $\mathcal H^n(A) = 0$. Is there any $A \subset \mathbb R^n$ with $\mathcal H^n(A) = 0$ but $\dim_H(A) = n$? Thank you.

• $\mathcal H^n$ means the Lebesgue measure, doesn't it? Sep 20, 2012 at 14:02
• $n$-dimensional Hausdorff measure. Yes it's equal to $n$-dimensional Lebesgue measure (with some scaling).
– Deco
Sep 20, 2012 at 14:06
• The Hausdorff measure $H^d(X)$ with $d=\dim_H(X)$ can be finite or zero or infinite. (As far as I know) Sep 20, 2012 at 14:34
• Intuitively, it's hard to imagine a set of measure zero has "full" dimension. But, a proof is still needed.
– Deco
Sep 21, 2012 at 1:46
• See a previous answer... math.stackexchange.com/a/132485/442 Sep 22, 2012 at 2:18

You can get a concrete example of such a set for the case $n=1$ by making a slight modification to the construction of the ternary Cantor set. Instead of taking away $\frac13$ of every interval at each step, remove $\frac{1}{k+1}$ in step $k$.

We can calculate the $d$-dimensional Hausdorff measure as follows. After $k$ steps we are left with $2^k$ intervals with a total length of $$\frac12 \cdot \frac23 \cdot \frac34 \cdots \frac{k}{k+1} = \frac{1}{k+1}$$ With some hand-waving we can then find the measure as $$\mathcal{H}^d(A) = \lim_{k\to\infty} 2^k \left(\frac{1}{(k+1)2^k}\right)^d = \lim_{k\to\infty} 2^{(1-d)k}(k+1)^{-d}$$ It is clear that for $1-d > 0$ the exponential beats the power function, making the measure infinite, but at $d=1$ the limit is $0$.

Note that this doesn't mean you can't get a nontrivial Hausdorff measure on this set. You just need to use a test function other than $x^d$. In this case I think $-x \log_2 x$ will work.
user
The text passage discusses the Hausdorff dimension and measure of subsets of $\mathbb{R}^n$, specifically asking if there can be a set with Hausdorff dimension equal to $n$ but with $n$-dimensional Hausdorff measure zero. It also provides an example of such a set for $n=1$ by modifying the construction of the Cantor set.

TYPE: SINGLE_QUESTION_WITH_ANSWER