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Yuval Peres
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In order to conclude the box dimension of the graph of $f$ is at most $2-\alpha$, it suffices to assume that the exceptional set (where the local $\alpha$-H"older condition fails to hold) has box dimension at most $1-\alpha$. This condition is sharp, unless you make further assumptions on the behavior of $f$ on the exceptional set.

Yuval Peres
  • 14.2k
  • 1
  • 28
  • 49