This is an answer to the first question in your comment above. (For notational convenience only I'll assume the Riemann Hypothesis.) You need to be careful regarding $\sim$. Since $\gamma_n\sim \frac{2\pi n}{\log n}$, $\gamma_n\sim\gamma_{n+1}$ is true. I claim $g_n\sim \frac{2\pi n}{\log n}$ as well: With $N_g(T)$ the number of Gram points to height $T$, one has
$$
N_g(T)=\frac{T}{2\pi}\log T+O(T)
$$
see Lemma (2.1) in the recent paper by Trudgian. So for some $c>0$,
$$
\frac{T}{2\pi}\log T-cT<N_g(T)<\frac{T}{2\pi}\log T+cT.
$$
Solving
$\frac{T}{2\pi}\log T-cT=n$ for $T$ gives that
$$
T=\frac{2\pi n}{W(2\pi n\exp(-2\pi c))}\sim \frac{2\pi n}{\log(n)},
$$
where $W(x)$ is the Lambert $W$-function, ProductLog in *Mathematica* (and *Mathematica* helps with the limits.) Similarly for the $\frac{T}{2\pi}\log T+cT$ term. So solving $N_g(T)=n$ for $T$ with $T=g_n$ gives the same asymptotic.

So $\gamma_n\sim g_n\sim \gamma_{n+1}\sim \frac{\gamma_n+\gamma_{n+1}}{2}$ trivially.