I am trying to find sets $X$ and $Y$ s.t. $\dim_B(X\bigcup{}Y)>\max\{\dim_B(X),\dim_B(Y)\}$.
At first I thought taking $X=[0,1]$ and $Y=\{10+1/n^2:n\ge{}1\}$ but I don't think that works. Is this even possible?
2026-02-23 19:24:02.1771874642
Are there two sets $X$ and $Y$ such that the following inequality for box dimension holds
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As Jens has mentioned that cannot be done for box dimension i.e. when the lower and the upper box dimensions coincide. But if you are talking about the lower box counting dimension, then there are such sets. You can check example 6.2 in Pesin's book: Dimension Theory in Dynamical Systems.