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Simplyorange
Simplyorange
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If $A\subset\mathbb{R}^2$ is a Borel measurable set and $p_\theta$ is projection onto the line spanned by $(\cos\theta,\sin\theta)$, then it is well known that for almost every $\theta\in[0,2\pi]$, $p_\theta(A)$ has Hausdroff dimension the min of 1 and the Hausdorff dimension of $A$.

My question is: let's say $A$ has Hausdorff dimension 1, what would be an example where $p_\theta(A)$ has Hausdorff dimension <1 for at least 3 different $\theta\in[0,2\pi]$? What if the number 3 is replaced by countable infinity? Or maybe such examples don't exist?

If $A\subset\mathbb{R}^2$ is a Borel measurable set and $p_\theta$ is projection onto the line spanned by $(\cos\theta,\sin\theta)$, then it is well known that for almost every $\theta\in[0,2\pi]$, $p_\theta(A)$ has Hausdroff dimension the min of 1 and the Hausdorff dimension of $A$.

My question is: let's say $A$ has Hausdorff dimension 1, what would be an example where $p_\theta(A)$ has Hausdorff dimension <1 for at least 3 different $\theta\in[0,2\pi]$? What if the number 3 is replaced by countable infinity? Or maybe such examples don't exist?

If $A\subset\mathbb{R}^2$ is a Borel measurable set and $p_\theta$ is projection onto the line spanned by $(\cos\theta,\sin\theta)$, then it is well known that for almost every $\theta\in[0,2\pi]$, $p_\theta(A)$ has Hausdroff dimension the min of 1 and the Hausdorff dimension of $A$.

My question is: let's say $A$ has Hausdorff dimension 1, what would be an example where $p_\theta(A)$ has Hausdorff dimension <1 for at least 3 different $\theta\in[0,2\pi]$? What if the number 3 is replaced by countable infinity?

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Simplyorange
Simplyorange
  • 265
  • 1
  • 5

Exceptional set for Marstrand's projection theorem

If $A\subset\mathbb{R}^2$ is a Borel measurable set and $p_\theta$ is projection onto the line spanned by $(\cos\theta,\sin\theta)$, then it is well known that for almost every $\theta\in[0,2\pi]$, $p_\theta(A)$ has Hausdroff dimension the min of 1 and the Hausdorff dimension of $A$.

My question is: let's say $A$ has Hausdorff dimension 1, what would be an example where $p_\theta(A)$ has Hausdorff dimension <1 for at least 3 different $\theta\in[0,2\pi]$? What if the number 3 is replaced by countable infinity? Or maybe such examples don't exist?