In Kamont "ON THE FRACTIONAL ANISOTROPIC WIENER FIELD" (found here : https://www.math.uni.wroc.pl/~pms/files/16.1/Article/16.1.6.pdf), on page 96, it is claimed that, if a function $f:I^{d}\rightarrow \mathbb{R}$ ($I$ being a real compact interval) verifies, for all $t,s \in I^{d}$, that, $$|f(t)-f(s)|=O(||t-s||^{\varepsilon})$$ then we get the following upper bound on the upper box counting dimension of the graph of $f$ (denoted $\Gamma(f)$), $$\overline{\operatorname{dim}}_{b}(\Gamma(f))\leq d+1-\varepsilon$$ For $d=1$, a proof of this claim can be found in Falconer, "FRACTAL GEOMETRY", but i haven't been able to find a proof of the general statement for all $d\geq 1$, nor to prove it myself. I am pretty sure this is standard, is there any canonical references I can look at and cite in order to use this result ?
1 Answer
For any $z\in\frac{\delta}{2}\mathbb{Z}^d\cap I^d$, put a box with side $\delta$ centered at each $(z,f(z)),(z,f(z)\pm \frac{\delta}{2}),\dots,(z,f(z)\pm N\cdot\frac{\delta}{2})$, where $N$ is the smallest integer such that $N\frac{\delta}{2}>C\left(\frac{\sqrt{d}\delta}{2}\right)^\varepsilon$ and $C$ is the constant in your $O(\|t-s\|^\varepsilon).$ For every point $w\in I^d$, the nearest point $z\in\frac{\delta}{2}\mathbb{Z}^d\cap I^d$ satisfies $w\in \left[-\frac{\delta}{2}, \frac{\delta}{2}\right]^d+z$ and $\|w-z\|\leq \frac{\sqrt{d}\delta}{2}$, and hence $(w,f(w))$ is covered by one of the boxes stacked over $z$. The number of boxes stacked over each $z$ is $O(\delta^{\varepsilon-1})$, and $|\frac{\delta}{2}\mathbb{Z}^d\cap I^d|=O(\delta^{-d})$, so, the total number of boxes in the cover is $O(\delta^{\varepsilon-1-d}).$
