= Solution
For <statistical homogeneity> and <statistical isotropy>, with the paper's finite-volume Fourier normalization,
$$
\xi(r)=\frac{V}{(2\pi)^3}\int d^3k\,P(k)e^{i\mathbf k\cdot\mathbf r}
=\frac{V}{2\pi^2}\int_0^\infty k^2P(k)\frac{\sin kr}{kr}\,dk.
$$
The angular integral is $4\pi\sin(kr)/(kr)$. In particular $\xi(r)=\int\Delta^2(k)\sin(kr)/(kr)\,d\ln k$, confirming the stated <dimensionless power spectrum> normalization.
For the <scale-free matter power spectrum> $P(k)=Ck^n$, set $u=kr$ to obtain
$$
\xi(r)=\frac{VC}{2\pi^2}r^{-n-3}I_n,
\qquad I_n=\int_0^\infty u^{n+1}\sin u\,du.
$$
At zero the integrand behaves as $u^{n+2}$, requiring $n>-3$; at infinity its oscillatory integral converges for $n<-1$. Thus in the ordinary convergent range $-3<n<-1$, the <scale-free density correlation transform> has
$$
I_n=\Gamma(n+2)\sin\frac{\pi(n+2)}2>0,
\qquad I_{-2}=\frac\pi2,
\qquad
\boxed{\gamma=n+3,\quad r_0^{\gamma}=\frac{VCI_n}{2\pi^2}.}
$$
The <gamma function> expression follows by introducing $e^{-\epsilon u}$, evaluating the complex Laplace integral where it converges absolutely at zero, and continuing within the convergent oscillatory range; its removable singularity at $n=-2$ is handled by the limit. Absolute convergence of the original correlation integral requires the smaller range $-3<n<-2$.
The exponent relation is a useful scaling statement beyond that interval, but a pure primordial $n\simeq1$ spectrum over all wavenumbers has no ordinary unsmoothed Fourier correlation integral. Physical transfer functions, cutoffs or smoothing must be specified, and a distributional extension need not yield a positive $r_0$ power law. Even within the convergent interval, the point variance is ultraviolet divergent without smoothing; the finite result here is at nonzero separation.
For a representative present-day ordinary galaxy sample over roughly $0.1$–$10\,h^{-1}\,\mathrm{Mpc}$, the familiar approximate fit is
$$
\boxed{r_0\simeq5\,h^{-1}\,\mathrm{Mpc},\qquad\gamma\simeq1.8,}
$$
where $H_0=100h\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$. These are empirical galaxy clustering parameters, not universal matter-spectrum constants; luminosity, color, selection and scale change the fit, as shown in https://arxiv.org/abs/astro-ph/0408569[the galaxy correlation measurements]. A formal power law of that slope corresponds to $n\simeq-1.2$ on those scales, not to the nearly scale-invariant primordial spectral index.
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