The normalized algorithmic information distance can not be approximated

It is known that the normalized algorithmic information distance $N$ is not computable and not semicomputable. We show that for all $ϵ<1\mathrm{/}2$, there exist no semicomputable functions that differ from $N$ by at most~$ϵ$. Moreover, for any computable function $f$ such that $|\lim_t f(x,y,t) - N(x,y)| \le \epsilon$andforall$n$,thereexiststrings$x,y$oflength$n$ such that ${\sum }_t\mathrm{\mid }f(x,y,t+1)-f(x,y,t)\mathrm{\mid }\ge \mathrm{\Omega }(\log n)$. This is optimal up to constant factors. We also show that the maximal number of oscillations of a limit approximation of $N$ is $\mathrm{\Omega }(n\mathrm{/}\log n)$. This strengthens the $\omega (1)$ lower bound from [K. Ambos-Spies, W. Merkle, and S.A. Terwijn, 2019, Normalized information distance and the oscillation hierarchy], see arXiv:1708.03583 .