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 $\Delta$-function by $\Delta(n) := \max_t \# \{d | n, \log d \in [t,t+1]\}$, we show that $\Delta(n) \geq (\log \log n)^{0.35332277\dots}$ for almost all $n$, 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 $A \subset \mathbb{N}$ by selecting $i$ to lie in $A$ with probability $1/i$. What is the supremum of all exponents $\beta_k$ such that, almost surely as $D \rightarrow \infty$, some integer is the sum of elements of $A \cap [D^{\beta_k}, D]$ 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.