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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.

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    $\begingroup$ If you take $i=0,j=N$ then the inequality becomes $C/N^{a-2}\le 1/N^a$, i.e $C \le N^{-2}$. Therefore I doubt if $C$ can be independent of all the values of $i,j,N$. $\endgroup$
    – Ralph
    Commented May 18, 2012 at 7:51
  • $\begingroup$ Thanks for the observation Ralph, you right such lower bound can not exists. $\endgroup$
    – Leandro
    Commented May 18, 2012 at 8:58
  • $\begingroup$ @Ralph: why not augment your comment into an answer so that the question can be officially laid to rest? $\endgroup$
    – Suvrit
    Commented May 18, 2012 at 15:44

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