Example of a sequence with certain decay properties

Question

Give an example for a sequence $a:\mathbb{Z}^n\to\mathbb{R}_{\geq1}$ which is:

1. Bounded.
2. Obeys the estimate $|\frac{a(k)}{a(m)}|\leq(1+\alpha\|k-m\|)^{-\beta}$ for all $k,m\in\mathbb{Z}^n$ for some given fixed $\alpha>0$ and $\beta>0$, where $\|\cdot\|$ is the Euclidean norm on $\mathbb{R}^n$.

If such a sequence cannot exist, I would like to know why.

Answer 1

I am not familiar with the notation, but I understand that $\Bbb R_{\ge1}=[1,\infty)$. Choosing $m=0$ we get the inequality $$ a(k)\le(1+\alpha\,\|k\|)^{-\beta}\,a(0)\quad\forall k\in\Bbb Z^n, $$ which is imposible since $a(k)\ge1$ for all $k\in\Bbb Z^n$.

What happens if $a(k)>0$ for all $k\in\Bbb Z^n$? Then, choosing $k=0$, we obtain $$ a(m)\ge(1+\alpha\,\|m\|)^{\beta}\,a(0)\quad\forall m\in\Bbb Z^n. $$ Since the sequence is bounded, this is again impossible.