Can there be a set whose Hausdorff dimension gradually changes?


For instance, a set of real numbers contained in an interval, whose Hausdorff dimension is 0 at the beginning and 1 closer to the end, and changes without jumps?



A set whose Hausdorff dimension gradually changes?
I assume you want a set $A\subseteq [0,1]$ such that $\dim (A\cap [0,x])=x$ for all $x$. We can define $A\_1$ by taking the union of a (Borel) subset of dimension $0$ of $[0,1/2]$ with a subset of dimension $1/2$ of $[1/2,1]$.To obtain $A\_2$, we then make our sets larger on $[1/4,1/2]$, $[3/4,1]$ and again, we make the dimension on each current interval equal to its left endpoint. Then $A\_1\subseteq A\_2\subseteq \ldots$, and $A=\bigcup A\_n$ works: by monotone convergence, $h^d(I\cap A)=\lim h^d(I\cap A\_n)$, so $A\cap [0,x]$ has the right dimension at each $x=k2^{-n}$ and thus everywhere.


