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Sorry, there was a typo. It should be $t(n)$, not $k(n)$. So the problem is given $s\ge 2$ (a fixed constant) and $t$ (which can be a function depending on $n$ and we can assume $t(n)\gg 1$), what can we say about $a$ and $b$?
Yes, $\tilde{\theta}(n)$ means it has order of magnitude of $n$ up to some power of logarithm factor $\log n$. For example $(\log n)^8 n$. While I think it is more meaningful to require $\tilde{\theta}(\sqrt{n})$, so I revised it.
