Hausdorff dimension of the graph of the sum of two continuous functions How can one prove the following result on the Hausdorff dimension of the graph of the sum of two continuous functions:

Let $f,g:[0,1] \to \mathbb R$ be two continuous functions. Suppose that $$\operatorname{dim}_H(\operatorname{graph} f) < \operatorname{dim}_H(\operatorname{graph} g). $$
Then $$\operatorname{dim}_H(\operatorname{graph} (g+f)) = \operatorname{dim}_H(\operatorname{graph} g)$$

 A: OK, it looks like I, indeed, need to spell a few things out.
First, continuous functions with Hausdorff dimension of the graph greater than $1$ exist. I'll skip this part.
Let $g$ be any such function. It can be written as the sum of a uniformly convergent series $\sum_k g_k$ of Lipschitz functions such that $\|g_k\|_\infty\le 2^{-k}$. If it is not already constructed like that, then just consider any sequence of piecewise linear approximations $G_k$ such that $\|G_k-g\|\le 2^{-k-2}$ and put $g_1=G_1$, $g_k=G_k-G_{k-1}$ for $k\ge 2$. Of course, the Lipschitz constants $L_k$ of $g_k$ will grow pretty fast.
Now let the first block be just $\{1\}$. Suppose that we have already constructed the blocks $J_1,\dots,J_n$ and $N$ is the last index of $J_n$. Let $L=\sum_{k=1}^N L_k$. Then the sum of $g_k$ over any subset of the union of the first $n$ blocks (which is just $[1,N]$) is $L$-Lipschitz, so for every $\delta>0$, we can cover the graph of that sum by about $L\delta^{-1}$ disks of radius $\delta$. Choose $\delta_n\in(0,2^{-n})$ so small that $L\delta_n^{-1}\le \delta_n^{-p_n}$ where $p_n\in(1,2)$ is some fixed sequence tending to $1$. Now let the $n+1$-st block to be $J_{n+1}=[N,M]$ where $M$ satisfies $2^{-M}\le\delta_n$. Then the graph of the sum of $g_k$ over any subset of indices disjoint with $J_{n+1}$ can be covered by $\delta_n^{-p_n}$ balls of radius $3\delta_n$, say (the tail beyond $M$ is just too small to really matter on that scale). 
Now, by construction, the sum of $g_k$ over odd blocks $J_n$ has a graph that can be covered by $\delta_n^{-p_n}$ balls of radius $3\delta_n$ for all odd $n$. Since $\delta_n\to 0$ and $p_n\to 1$, we conclude that its Hausdorff (or even lower box) dimension is $1$. The same is true for the sum over even blocks. 
A: An explicit counterexample is in Corollary 1.6 of 
Peres, Yuval, and Perla Sousi. "Dimension of fractional Brownian motion with variable drift." Probability Theory and Related Fields 165, no. 3-4 (2016): 771-794.
https://arxiv.org/pdf/1310.7002.pdf
