Density results for $r$-gaps between zeros of the Riemann zeta-function

Let $0<{\gamma }_1\le {\gamma }_2\le \dots$ denote the positive ordinates of the non-trivial zeros of the Riemann zeta-function. A result first announced by Selberg states that there exist absolute constants $\mathrm{\Theta },\vartheta >0$ such that for each $r\in \mathbb{N}$, $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{{\gamma }_{n+r}-{\gamma }_n}{2\pi r\mathrm{/}\log {\gamma }_n}\ge 1+\frac{\mathrm{\Theta }}{r^{\alpha }}\text{and}\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{{\gamma }_{n+r}-{\gamma }_n}{2\pi r\mathrm{/}\log {\gamma }_n}\le 1-\frac{\vartheta }{r^{\alpha }}$$ where $\alpha$ may be taken as $2\mathrm{/}3$, or as $1\mathrm{/}2$ if one assumes the Riemann hypothesis. This was recently proved by Conrey and Turnage-Butterbaugh under RH and by Inoue unconditionally. We prove that in fact a positive proportion of $r$-gaps are large (and small) to the above extent, and we provide explicit estimates for the sizes and proportions of these gaps. In the case $r=1$, this quantitatively improves an unconditional result of Simonič, Trudgian and Turnage-Butterbaugh.