ScalingStacks

Proof. For any $M\leq m\leq 2M-1$, we have $t_{1}^{\otimes m}\in V_{m}(L_{X|Y})$. By Corollary 3.27, for any $\epsilon>0$, there exists $N_{m}\in\mathbb{N}$ such that for $l\geq N_{m}$,

$$\vvvert t_{1}^{\otimes ml}\vvvert_{\phi,X|Y}^{\frac{1}{l}}/\vvvert t_{1}^{\otimes m}\vvvert_{\phi|_{Y}}\leq\epsilon.$$

It is easy to see that there exists $n_{Y}\in\mathbb{N}$ such that the set of integers

$$\{ml:l\geq N_{m},M\leq m\leq 2M-1\}$$

contains a subset of form $\mathbb{N}-\{0,\dots,n_{Y}-1\}$: the case $M=1$ is clear; if $M\geq 2$, the fact that $M$ and $M+1$ are coprime guarantees the existence of $n_{Y}$. Note that $\vvvert\mathord{\cdot}\vvvert_{\phi|_{Y}}$ is power-multiplicative, so for any $n\geq n_{Y}$, there exists $s_{n}\in V_{n}(L)$ with $s_{n}|_{Y}=t_{1}^{\otimes n}$ such that

$$\lVert s_{n}\rVert_{n\phi}\leq\mathrm{e}^{n\epsilon}\cdot\lVert t_{1}^{\otimes n}\rVert_{n\phi|_{Y}}=\mathrm{e}^{n\epsilon}\cdot(\lVert t_{1}\rVert_{\phi|_{Y}})^{n}.$$

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