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



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


