Suppose we have $X, Y \subset \mathbb{R}^n $ then $\max \{\dim X, \dim Y \}\leq \dim (X\cup Y) $ where $\dim $ means lower box counting dimension.
What examples are there of of this inequality being strict? Can't find any resources or anything online about stuff like this. Any sources or books or anything would be helpful if you know any.
I do not know if this helps you much, but you might want to take a look at the book "Fractal Geometry. Mathematical Foundations and Applications" by Kenneth Falconer. There, your problem appears as Exercise 3.10 (2nd edition of the book) and there is at least a hint how to solve it (by refering you to Exercise 3.8).