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Question: Is there a shift-invariant submultiplicative seminorm $||\cdot||$ of $\ell^\infty$ which satisfies the following property?

If $f:\mathbb{N}\rightarrow\mathbb{N}$ is an increasing function bounded below by some exponential function $cr^n$ for real numbers $c>0,r>1$, then the (bounded) sequence $||(\frac{f(0)}{f(1)},\frac{f(1)}{f(2)},\frac{f(2)}{f(3)},\dots)||<1$

More generally, if $f,g$ are two increasing functions on $\mathbb{N}$ such that $g(n)\le f(n)$ then $||(\frac{f(0)}{f(1)},\frac{f(1)}{f(2)},\frac{f(2)}{f(3)},\dots)||<||(\frac{g(0)}{g(1)},\frac{g(1)}{g(2)},\frac{g(2)}{g(3)},\dots)||$

(This, for example, excludes $\lim\sup$, if we take $f(n)=4^{\lfloor\frac{n}{2}\rfloor+1}$ and $g(n)=2^n$)


This comes up when estimating the asymptotic growth of certain "sparse" infinite binary trees. If there were such a norm, then the number of nodes at depth $n$ in such trees could not be bounded below by any such exponential function. By the way, I have no knowledge of functional analysis.

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