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2. Positive results

    1.     For a set $E$ in a metric space, define $\Delta(E) \subseteq \R$ by \[ \Delta(E) = \{d(x, y) \mid x, y \in E\}. \]

      Add remark

      Problem 2.1.

      [Don Stull] For what $s$ can we prove that if $E \subseteq \R^2$ is of Hausdorff dimension greater than $1$ then $\text{dim}_H(\Delta(E)) \geq s$?
          Stull has recently proved that this is possible for $s = 3/4$, improving the best previous result of $\sim 0.7$. It is conjectured that $s = 1$.


        • Add remark

          Problem 2.2.

          [Dan Turetsky] Marstrand’s theorem for works when $\text{dim}_H(E) = \text{dim}_P(E)$—i.e. for such an $E$ and for almost every direction $e$, $\text{dim}_H(\text{proj}_e(E)) = \min(1, \text{dim}_H(E))$. What can we say if we instead assume $\text{dim}_P(E) \leq \text{dim}_H(E) + \varepsilon$?
              One way to address this question might be to come up with an “up-to-$\varepsilon$” version of Hausdorff optimal oracles.

            •     Marstrand’s second projection theorem states that if $E \subseteq \R^2$ is an analytic set such that $\text{dim}_H(E) > 1$ then for almost every direction $e$, $\text{dim}_H(\text{proj}_e(E))$ has positive Lebesgue measure.

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              Problem 2.3.

              [Neil Lutz] Is there an effective proof of Marstrand’s second projection theorem? Also, is the following effective version of the theorem true: if $x \in \R^2$ is such that $\text{dim}(x) > 1$ and $e$ is Martin-Löf random relative to $x$ then is $\text{proj}_e(x)$ Martin-Löf random (or, better yet, random relative to $e$)?

                •     The following question is an attempt to quantify the randomness required by Marstrand’s second projection theorem.

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                  Problem 2.4.

                  [Don Stull] Is there a set $E \subseteq [0,1]^2$ which is $\Pi^0_1$ and of Hausdorff dimension $> 1$ such that for every non-Schnorr random direction $e$, $\text{proj}_e(E)$ has measure $0$? Or replace Schnorr randomness with other randomness notions.


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                      Problem 2.5.

                      [Elvira Mayordomo] Are there geometric properties that imply (or are equivalent to) a set having a Hausdorff optimal oracle? More generally, is the definition of Hausdorff optimal oracle “right”? Is there a better definition?


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                          Problem 2.6.

                          [Alexander Shen] Are there any known inequalities for Kolmogorov complexity that imply interesting results in geometric measure theory via the point-to-set principle?

                            •     Fix an enumeration $f \colon 2^{< \omega} \to \Q_2$ (where $\Q_2$ indicates dyadic rationals) and $x \in \R$ and look at \[ \text{dim}_{FS}^f(x) = \inf_{A \in \text{finite state machines}}\liminf_{r \to \infty} K^f_{A, r}(x)/r \] Say $x$ is $f$-normal if $\text{dim}_{FS}^f(x) = 1$. If $f$ is standard enumeration of dyadic rationals, we get $2$-normality.

                              Add remark

                              Problem 2.7.

                              [Elvira Mayordomo] For which $f$ do we have $(2^k\cdot x)_{k \in \N}$ equidistributed mod $1$ whenever $x$ is $f$-normal?

                                  Cite this as: AimPL: Effective methods in measure and dimension, available at http://aimpl.org/algorandom.