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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?

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    $\begingroup$ I presume you mean $\max\{\dim_B(X), \dim_B(Y)\}$. Such sets do not exist as box counting dimension is finitely additive. $\endgroup$ – Jens Renders May 22 at 18:12
  • $\begingroup$ Surely, if its finitely additive then $\dim_B(X\cup{}Y)=\dim_B(X)+\dim_B(Y)>\max{(\dim_B(X),\dim_B(Y))}$? $\endgroup$ – kam May 22 at 18:34
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    $\begingroup$ No, it means it is equal to the maximum. Take a look at this $\endgroup$ – Jens Renders May 22 at 19:08
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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.

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