# Are there two sets $X$ and $Y$ such that the following inequality for box dimension holds

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?

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