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 Δ (n) := max _t #{d | n, log d ∈ [t,t+1]} , we show that Δ (n) ⩾ (loglog n)^0.35332277… 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⊂ℕ by selecting i to lie in A with probability 1/ i . What is the supremum of all exponents β _k such that, almost surely as D →∞ , some integer is the sum of elements of A∩ [D^β _k, D] in k different ways? We characterise β _k as the solution to a certain optimisation problem over measures on the discrete cube {0,1}^k , and obtain lower bounds for β _k which we believe to be asymptotically sharp.