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1. Pathological sets

    1.     P. Lutz and J. Miller have shown that, assuming CH, there is some set $E \subseteq \R^2$ such that $\text{dim}_H(E) = 1$ and for any continuous function $f\colon \R^2 \to \R$, $\text{dim}_H(f(E)) = 0$. This is a strengthening of a theorem of Davies, which has the same form but applies only to linear projections $\R^2 \to \R$ rather than to all continuous functions.

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      Problem 1.1.

      [Joe Miller] Assume $CH$ and fix $s \in [0, 1)$. Is there a set $E \subseteq \R^2$ such that $\text{dim}_H(E) = 1 + s$ and for every continuous function $f \colon \R^2 \to \R$, $\text{dim}_H(f(E)) \leq s$?
          Note that such a result must use some topological properties of $\R^2$ because the analogous result is false for $2^\omega$—there is a continuous function $f \colon 2^\omega\times 2^\omega \to 2^\omega$ which sends any set of Hausdorff dimension $d$ to a set of Hausdorff dimension $d/2$.

        •     P. Lutz and J. Miller have also shown the following theorem. Assume CH and fix $s \in [0, 1]$. Then there a set $E \subseteq 2^\omega$ such that $\text{dim}_H(E) = s$ and for every continuous $f \colon 2^\omega \to 2^\omega$, $\text{dim}_H(f(E)) \leq s$.

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          Problem 1.2.

          [org.aimpl.user:pglutz@math.ucla.edu] Assume CH and fix $s \in [0, 1]$. Is there a set $E \subseteq 2^\omega$ such that $\text{dim}_H(E) = s$ and for every Borel $f \colon 2^\omega \to 2^\omega$, $\text{dim}_H(f(E)) \leq s$?
              It would be interesting to find any nontrivial upper bound on the complexity of $f$.


            • Add remark

              Problem 1.3.

              [org.aimpl.user:pglutz@math.ucla.edu] Let $Z \in 2^\omega$ be arbitrary and fix $s \in (0, 1)$. Is there some $X \in 2^\omega$ such that $\text{dim}^Z(X) = s$ and for all $Y \leq_T X'$, $\text{dim}^{0'}(Y) \leq s$?
                  It seems likely that if this question has a positive answer then so does the previous one.

                •     Besicovitch showed that, assuming CH, there is a set $E \subseteq \R^2$ such that $E$ is not $\sigma$-finite in dimension $1$ but for every superlinear gauge function $\varphi$, $H_\varphi(E) = 0$. One can also get such a set $E$ from a counterexample to the Borel conjecture (i.e. the statement that every strong measure $0$ set is countable).

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                  Problem 1.4.

                  [Ted Slaman] Does the existence of a set $E$ such that $E$ is not $\sigma$-finite in dimension $1$ but for every superlinear gauge function $\varphi$, $H_\varphi(E) = 0$ imply that there is a counterexample to the Borel conjecture?


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                      Problem 1.5.

                      [Denis Hirschfeldt] Most constructions of sets with pathological behavior related to Hausdorff dimension require some assumption beyond ZFC, i.e. CH, Martin’s axiom, the failure of the Borel conjecture, etc. Is it provable in ZFC that such pathological sets exist or is their existence independent of ZFC? For a concrete example, is it provable in ZFC that there is a set without a Hausdorff optimal oracle?

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