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I have been thinking about a couple different problems in fractal geometry (including I one deleted because it was ill posed) and realize they all depend in a fundamental way on the problem of: Can we ...

Let $f:\mathbb{R}^d\rightarrow\mathbb{R}$ be a continuous function and $\mathrm{supp}(f) := \mathrm{cl}\{x\in\mathbb{R}^d\mid f(x)\neq 0\}$ its support. Under which conditions is it true that $f≠0$ (...