Equal sums in random sets and the concentration of divisors

Abstract

AbstractWe study the extent to which divisors of a typical integer n are concentrated. In particular, defining $$\Delta (n) := \max _t \# \{d | n, \log d \in [t,t+1]\}$$ Δ ( n ) : = max t # { d | n , log d ∈ [ t , t + 1 ] } , we show that $$\Delta (n) \geqslant (\log \log n)^{0.35332277\ldots }$$ Δ ( n ) ⩾ ( log log 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 $${\textbf{A}} \subset {\mathbb {N}}$$ A ⊂ N by selecting i to lie in $${\textbf{A}}$$ A with probability 1/i. What is the supremum of all exponents $$\beta _k$$ β k such that, almost surely as $$D \rightarrow \infty $$ D → ∞ , some integer is the sum of elements of $${\textbf{A}} \cap [D^{\beta _k}, D]$$ A ∩ [ D β k , D ] in k different ways? We characterise $$\beta _k$$ β k as the solution to a certain optimisation problem over measures on the discrete cube $$\{0,1\}^k$$ { 0 , 1 } k , and obtain lower bounds for $$\beta _k$$ β k which we believe to be asymptotically sharp.