Bertrand's Postulate for Carmichael Numbers

Alford, Granville, and Pomerance proved that there are infinitely many Carmichael numbers. In the same paper, they ask if a statement analogous to Bertrand’s postulate could be proven for Carmichael numbers. In this paper, we answer this question, proving the stronger statement that for all $\delta >0$ and $x$ sufficiently large in terms of $\delta$, there exist at least $e^{\frac{\log x}{(\log \log x{)}^{2+\delta }}}$ Carmichael numbers between $x$ and $x+\frac{x}{(\log x{)}^{\frac{1}{2+\delta }}}$.