I have this bound for the sum
\begin{align} g(t,\eta) &\le A D^{(K-d)/2} \sup_{x \in \mathbb{R}^d} \sum_{m \in \mathbb{Z}^d, |x-m|>t}|x-m|^{-K} \\ & =A D^{(K-d)/2} \sup_{x \in \mathbb{R}^d} \sum_{j=0}^{\infty} \sum_{m \in \mathbb{Z}^d,t+j<|x-m|<t+j+1}|x-m|^{-K}\end{align}
Now the author writes: since the number of integers $m \in \mathbb{Z}^d$, for which $j \le |x-m| \le j+1$ is bounded by $C \cdot (j)^{d-1}$
\begin{align} g(t,\eta) &\le A D^{(K-d)/2}C \sum_{j=0}^{\infty}|t+j|^{d-1-K}\end{align}
I don't understand how the author write:
since the number of integers $m \in \mathbb{Z}^d$, for which $j \le |x-m| \le j+1$ is bounded by $C \cdot (j)^{d-1}$. I want an expression of this constant $C>0$. Is it possible?