This question is related to a work in progress about Ballistic-Diffusive phase transition for some random polymers with long range self-repulsion.
The motivation to ask here if the inequality below holds is to avoid a length analysis in this paper.
Let $0\leq i<j\leq N$ be integers and $\alpha \in (3,4]$ is there a constant $C>0$ independent of $i,j$ and $N$ such that
$$
\frac{C}{|i-j|^{\alpha-2}}\leq \sum_{0\leq k\leq i< j\leq r \leq N}
\frac{1}{|k-r|^{\alpha}}
$$
holds ?
Remark: similar upper bound holds, to get it I simply compared this sum with a double integral. For the lower bound it seems that such comparison does not work.
