Power tempering bounds interactions between separated cubes

Let $m=\mathrm{\parallel }j-j^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}\ge 1$ for two distinct cube indices. In a coordinate where the index difference is $m$, the nearest points of the two intervals are separated by $m(\mathrm{\ell }+g)-\mathrm{\ell }=(m-1)\mathrm{\ell }+mg\ge mg$. Hence every pair of points in $C_j,C_{j^{\mathrm{\prime }}}$ has Euclidean distance at least $mg$.

For $m\ge 1$, [F3] gives the shell count. Use $x^d-y^d=(x-y){\sum }_{r=0}^{d-1}{x}^{d-1-r}{y}^r$ with $x=2m+1$, $y=2m-1$. Since $2m+1\le 3m$, it yields $(2m+1{)}^d-(2m-1{)}^d\le 2d(3m{)}^{d-1}=2d{3}^{d-1}{m}^{d-1}\mathrm{.}$ Therefore $S_{d,ϵ}\le 2d{3}^{d-1}{\sum }_{m\ge 1}{m}^{-1-ϵ}<\mathrm{\infty }$ by [F2], since $1+ϵ>1$.

By [F1] and step 1.1, each such pair contributes at most $C(mg{)}^{-d-ϵ}$ in absolute value, because $mg\ge g\ge R$ and the Euclidean norm dominates the absolute value of any coordinate.

Fix $j\in J$. For each $j^{\mathrm{\prime }}\mathrm{\ne }j$, there are at most $n_j{n}_{j^{\mathrm{\prime }}}\le n^2$ labelled pairs of positions from $C_j$ and $C_{j^{\mathrm{\prime }}}$. Summing the bound from step 2.1 over those pairs and enlarging the finite set of index differences to all nonzero lattice vectors gives at most $C{n}^2{S}_{d,ϵ}{g}^{-d-ϵ}$, since [F3] groups those vectors into the shells whose counts define $S_{d,ϵ}$, and [F2] defines its nonnegative sum. Summing this bound over all $M$ cubes counts each unordered cube pair twice, so dividing by $2$ gives the claimed inequality.