Sums of square roots that are close to an integer

Abstract

Let k∈N and suppose we are given k integers 1≤a1,…,ak≤n. If a1+…+ak is not an integer, how close can it be to one? When k=1, the distance to the nearest integer is ≳n−1/2. Angluin-Eisenstat observed the bound ≳n−3/2 when k=2. We prove there is a universal c>0 such that, for all k≥2, there exists a ck>0 and k integers in {1,2,…,n} with0<‖a1+…+ak‖≤ck⋅n−c⋅k1/3, where ‖⋅‖ denotes the distance to the nearest integer. This is a case of the square-root sum problem in numerical analysis where the usual cancellation constructions do not apply: already for k=3, the problem appears hard.