On the intermediate values of the box dimensions

We address the following question: Is it true that, for every metric compactum $X$ of box dimension $\dim_BX=a\leq\infty$ and every two reals $\alpha$ and $\beta$ such that $0\leq\alpha\leq\beta\leq a$, there exists a closed subset in $X$ whose lower box dimension is $\alpha$ and whose upper box dimension is $\beta$? We give the positive answer for $\alpha=0$. In the general case, this result is final. We construct an example of a metric compactum whose box dimension is $1$ but every nonempty proper closed subset of the compactum has lower box dimension $0$.