Equal sums in random sets and the concentration of divisors

Abstract

We study the extent to which divisors of a typical integer n are concentrated. In particular, defining Delta (n) := max _t # {d | n, log d in [t,t+1]}, we show that Delta (n) geqslant (log log n)^{0.35332277ldots } 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 {textbf{A}} subset {mathbb {N}} by selecting i to lie in {textbf{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 {textbf{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.