Equal sums in random sets and the concentration of divisors

We study the extent to which divisors of a typical integer $n$ are concentrated. In particular, defining the Erd\H{o}s-Hooley $\mathrm{\Delta }$-function by $\mathrm{\Delta }(n):={\max }_t\mathrm{\#}\{d\mathrm{\mid }n,\log d\in [t,t+1]\}$, we show that $\Delta(n) \geq (\log \log n)^{0.35332277\dots}$foralmostall$n$, a bound we believe to be sharp. This disproves a conjecture of Maier and Tenenbaum. We also prove analogs for the concentration of divisors of a random permutation and of a random polynomial over a finite field. Most of the paper is devoted to a study of the following much more combinatorial problem of independent interest. Pick a random set $A \subset $\mathbb{N}$$byselecting$i$toliein$A$withprobability$1/i$. What is the supremum of all exponents ${\beta }_k$ such that, almost surely as $D \rightarrow \infty$,someintegeristhesumofelementsof$A \cap [D^{\beta_k}, D]$in$k$ different ways? We characterise ${\beta }_k$ as the solution to a certain optimisation problem over measures on the discrete cube $\{0,1{\}}^k$, and obtain lower bounds for ${\beta }_k$ which we believe to be asymptotically sharp.