interpretation of logarithmic in the dimension

Question

I'm reading a paper that says

"$\ Cn^{6/5}r \log n $ is not logarithmic or polylogarithmic in the dimension and one would like to know if results closer to the $\ nr \log n$ limit are possible."

Isn't it logarithmic in the dimension since the $\log n$ is there in both cases? (The limit isn't logarithmic in the dimension anyway so why bother pointing out that it isn't so in the first case?)

Answer 1

Isn't it logarithmic in the dimension since the $\log n$ is there in both cases?

The presence of $\log n$ does not make something logarithmic in $n$. The bound of the form $)(\log n)$ does. Since it is not true that $n^{6/5}\log n =O(\log n)$, the expression is not logarithmic. It is not polylogarithmic either, since there is no bound of the form $n^{6/5}\log n =O((\log n)^k)$ for any fixed $k$.

(To prove the latter claim, compare the logarithms of both sides: $\frac65 \log n + \log\log n$ is linear in $\log n$, while $k\log \log n$ grows much slower.)

The limit isn't logarithmic in the dimension anyway so why bother pointing out ...

Not clear to me either. Maybe this paragraph makes more sense in wider context.