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Consider a cube $Q\subset\mathbb R^n$ of side length $L$. Partition this into $m^n$ smaller cubes of side length $\frac L m$, and diameter $\frac{L\sqrt n}{m}$. Let $s > n$, and $\delta = \frac{L\sqrt n}{m}$. $$H^s_{\delta}(Q) \le m^n \left(\frac{L\sqrt n}{m}\right)^s = m^{n-s} L^s n^{s/2}$$
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Sun, 16 May 2021 17:22 GMT