Photon-box energy and singular entropy Riemann limits

For a continuous compactly supported $h:{\mathbb{R}}^3\to \mathbb{R}$, the cubes $Q_{n,d}=dn+[-d\mathrm{/}2,d\mathrm{/}2{]}^3$ tile ${\mathbb{R}}^3$ up to boundaries of volume zero. Uniform continuity of $h$ bounds $\mathrm{\mid }{d}^3{\sum }_{n}h(dn)-\int h\mathrm{\mid }$ by the volume of one fixed enlarged support box times the modulus of continuity at $\sqrt{3}d\mathrm{/}2$, which tends to zero. Taking $h(x)=\psi (\mathrm{\mid }x\mathrm{\mid })$ proves the compactly supported radial lattice limit. To rewrite its integral when $\psi$ is signed, apply [F2] separately to the nonnegative radial functions $h_{+}(x)=\max (h(x),0)$ and $h_{-}(x)=\max (-h(x),0)$; both are continuous and compactly supported, so their integrals are finite. Subtracting the resulting identities and using $\sigma (S^2)=4\pi$ from [F3] gives ${\int }_{{\mathbb{R}}^3}h(x)d^{3}x=4\pi {\int }_0^{\mathrm{\infty }}{r}^2\psi (r)dr$. Omitting $n=0$ changes the sum by $d^3\psi (0)\to 0$.

For $0<r\le 1$, $f(r)\le 1$ and $1-e^{-r}\ge r\mathrm{/}2$, so $0\le g(r)\le C(1+\mathrm{\mid }\log r\mathrm{\mid })$. Formulae [F2] and [F3] give the small-ball integral bound because, for $0<\delta \le 1$, ${\int }_0^{\delta }{r}^2(1+\mathrm{\mid }\log r\mathrm{\mid })dr={\delta }^3(4\mathrm{/}9+\mathrm{\mid }\log \delta \mathrm{\mid }\mathrm{/}3)$. For the discrete sum with $0<d\mathrm{\mid }n\mathrm{\mid }\le \delta$, put $j=\mathrm{\mid }n{\mathrm{\mid }}_{\mathrm{\infty }}$; then $dj\le \delta$ and $d\mathrm{\mid }n\mathrm{\mid }\ge dj$. Group the shells by $dj\in (\delta 2^{-\mathrm{\ell }-1},\delta 2^{-\mathrm{\ell }}]$. If such a group is nonempty, then $d\le \delta 2^{-\mathrm{\ell }}$ and its shell indices obey $j\le J_{\mathrm{\ell }}:=\lfloor \delta 2^{-\mathrm{\ell }}\mathrm{/}d\rfloor$. By [F1], the number of lattice points in shells up to $J_{\mathrm{\ell }}$ is ${\sum }_{j\le J_{\mathrm{\ell }}}(24j^2+2)\le C{J}_{\mathrm{\ell }}^3\le C(\delta 2^{-\mathrm{\ell }}\mathrm{/}d{)}^3$. Every point in the selected small-ball subset of this group has $d\mathrm{\mid }n\mathrm{\mid }\ge dj>\delta 2^{-\mathrm{\ell }-1}$ and $d\mathrm{\mid }n\mathrm{\mid }\le \delta \le 1$, so its entropy value is at most $C(1+\mathrm{\mid }\log \delta \mathrm{\mid }+\mathrm{\ell })$. Multiplication by the cell volume $d^3$ bounds this group's contribution by $C{\delta }^3{2}^{-3\mathrm{\ell }}(1+\mathrm{\mid }\log \delta \mathrm{\mid }+\mathrm{\ell })$. Summing the convergent geometric series over $\mathrm{\ell }\ge 0$ gives $C^{\mathrm{\prime }}{\delta }^3(1+\mathrm{\mid }\log \delta \mathrm{\mid })$ uniformly in $d$.

The function $f$ is continuous at zero since ${\lim }_{r\downarrow 0}r\mathrm{/}(e^r-1)=1$, and for $r\ge 1$ it is bounded by $C(1+r)e^{-r}$. For $0<d\le 1$ and $x\in Q_{n,d}$, $\mathrm{\mid }\mathrm{\mid }dn\mathrm{\mid }-\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }\le \sqrt{3}\mathrm{/}2$. Thus, for $R\ge 2$, each cube with $\mathrm{\mid }dn\mathrm{\mid }>R$ lies in $\mathrm{\mid }x\mathrm{\mid }>R-\sqrt{3}\mathrm{/}2>1$, and the exponential bound yields ${\sum }_{\mathrm{\mid }dn\mathrm{\mid }>R}{d}^{3}f(d\mathrm{\mid }n\mathrm{\mid })\le C^{\mathrm{\prime }}{\int }_{\mathrm{\mid }x\mathrm{\mid }>R-\sqrt{3}\mathrm{/}2}(1+\mathrm{\mid }x\mathrm{\mid })e^{-\mathrm{\mid }x\mathrm{\mid }}{d}^{3}x$. The integral tail tends to zero as $R\to \mathrm{\infty }$, uniformly in $0<d\le 1$. Apply step 1.1 on compact cutoffs and then this uniform tail bound to obtain the asserted $f$ limit and integrability.

For each fixed $d>0$, only finitely many lattice points have $d\mathrm{\mid }n\mathrm{\mid }<1$, and there are only finitely many shells with $dj<1$. On every remaining shell, $dj\ge 1$, the Euclidean norm satisfies $j\le \mathrm{\mid }n\mathrm{\mid }\le \sqrt{3}j$, and the exponential bound gives $f(d\mathrm{\mid }n\mathrm{\mid }),g(d\mathrm{\mid }n\mathrm{\mid })\le C(1+dj)e^{-dj}$. The shell count [F1] therefore bounds the shell contribution by $C{j}^2(1+dj)e^{-dj}\le C{j}^3(1+dj)e^{-dj}$. The comparison series converges by the ratio test, since the ratio of consecutive terms tends to $e^{-d}<1$. For $r\ge 1$, $-\log (1-e^{-r})\le e^{-r}\mathrm{/}(1-e^{-r})$, so $g(r)\le C(1+r)e^{-r}$; the cube estimate of step 2.1 then gives uniformly vanishing large-radius tails for the integral and the lattice sum. On each annulus $\delta \le \mathrm{\mid }x\mathrm{\mid }\le R$, $g(\mathrm{\mid }x\mathrm{\mid })$ is continuous and the compact Riemann-sum argument of step 1.1 applies with a continuous radial cutoff equal to one on that annulus and zero near the origin and outside a larger ball. Letting $\delta \downarrow 0$ and $R\to \mathrm{\infty }$ using the small-ball bound of step 1.2 and these tail bounds proves the $g$ limit and its finiteness.