I am new to fractals and dimension theory, so please excuse any errors in my understanding.
For a set $F$, let $dim_b (F)$ be the box counting dimension of $F$, and $dim_H (F)$ be the Hausdorff dimension of $F$.
I understand that, assuming the box counting dimension exists, $dim_H (F) \leq dim_b(F)$, and that this inequality is in general strict.
I have now come across the open set condition in Falconer's book, and my question is as follows:
When $F$ satisfies the open set condition,(regardless of any other properties of $F$) is this an example of when $dim_H(F) = dim_b(F)$?
Yes, you are correct. However, much more is true. If $F$ is a strictly self-similar set, then $\dim_H(F) = \dim_b(F)$. This is true regardless of whether the open set condition is satisfied. This surprising fact was proven in 1989 by Falconer; it appears as example 2 on page 550 in his paper "Dimensions and measures of quasi self-similar sets".
The importance of the open set condition is that it makes self-similar sets much more easy to analyze. For example, when OSC is satisfied, then the common value $d$ of the Hausdorff and box dimension is given by the well known Moran equation, namely $$r_1^d + r_2^d + \cdots + r_n^d = 1.$$ In the absence of the open set condition, this formula is no longer applicable and, indeed, the exact common value of the dimensions may not be computable via known methods.