[Solved]: Language with $loglog n$ space complexity?

I found the answer below in lecture notes of Muli Safra. Consider the language consisting of the following strings: $$ begin{align*} & 0 $ 1 $ & 00 $ 01 $ 10 $ 11 $ & 000 $ 001 $ 010 $ 011 $ 100 $ 101 $ 110 $ 111 $ &ldots end{align*} $$ This language can be recognized in space $O(loglog n)$. For each $m$ which is smaller than the width of the first block, we check that the $m$ least significant bits form a counter modulo $m$ starting at $0$ and ending at $2^m-1$. We stop when $m$ is the width of the first block. Since this language clearly isn’t regular, it requires space $loglog n$ (see for example these lecture notes by Hansen). We conclude that the language requires space $Theta(loglog n)$.