I've asked a couple questions about box dimensions (also called the Minkowski-Bouligand dimensions) recently. My main goal is to better understand the box dimension of discrete sets of points within the interval $[0,1]$; in particular, I'm interested in ways to compute the box dimension quickly. My hope is that the following question would give me a useful tool in computing box dimensions.
Suppose that we have two sets of points $f(n)$ and $g(n)$ taking values between $0$ and $1$. I imagine that $f$ and $g$ are monotonically decreasing functions with a non-negative second derivative, like $\frac{1}{n}$ and $e^{-n}$. Suppose that the box dimensions of these sets of points satisfy $B(\{f(n): n=1,2,3...\}) \geq B(\{g(n): n=1,2,3...\})$. Are we guaranteed that $B(\{f(n)+g(n): n=1,2,3...\}) = B(\{f(n): n=1,2,3...\})$? That is, if we "perturb" the points of $f(n)$ by $g(n)$, do we keep the same box dimension?
This question was motivated by a comment by Alex K - if it turns out that we are guaranteed the equality above, this could ease the computation of the box dimension of weakly perturbed sets of points.
