  We give an improved lower bound for the average of the Erd\H{o}s-Hooley
function $\Delta(n)$, namely $\sum_{n\le x} \Delta(n) \gg_\varepsilon
x(\log\log x)^{1+\eta-\varepsilon}$ for all $x\geqslant100$ and any fixed
$\varepsilon$, where $\eta = 0.3533227\dots$ is an exponent previously
appearing in work of Green and the first two authors. This improves on a
previous lower bound of $\gg x \log\log x$ of Hall and Tenenbaum, and can be
compared to the recent upper bound of $x (\log\log x)^{11/4}$ of the second and
third authors.
