Problem in computational complexity (superior class)

Question

Say that a class $C_1$ is superior to a class $C_2$ if there is a machine $M_1$ in class $C_1$ such that for every machine $M_2$ in class $C_2$ and every large enough $n$, there is an input of size between $n$ and $n^2$ on which $M_1$ and $M_2$ answer differently.

1. Is DTIME($n^{1.1}$) superior to DTIME($n$)?

2. Is NTIME($n^{1.1}$) superior to NTIME($n$)?

This is an exercise from book Computational Complexity: A Modern Approach. But I have no idea to handle it. Does it have relationship with theorem that DTIME$(n)$ $\subsetneqq$ DTIME($n^{1.5}$)?

Answer 1

The proof of the time hierarchy theorems can be paraphrased like this. We want to show that $\mathsf{X} \subsetneq \mathsf{Y}$ for appropriate $\mathsf{X},\mathsf{Y}$. The idea is to construct a diagonalizing machine $M \in \mathsf{Y}$ (slight abuse of notation here) whose language is different from all languages in $\mathsf{X}$. On input $x$, the machine $M$ runs $x$ on $x$ up to some specified time bound, and then answers the opposite. If the simulation is efficient enough, $M \in \mathsf{Y}$.

Your question is very similar. Go over the proof of the time hierarchy theorems, and check whether the diagonalizing machines there fit your bill.