Sums of square roots that are close to an integer

Abstract

Let $k\in N$ and suppose we are given k integers $1\le a_1,\dots ,a_k\le n$. If $\sqrt{a_1}+\dots +\sqrt{a_k}$ is not an integer, how close can it be to one? When $k=1$, the distance to the nearest integer is $\gtrsim n^{-1/2}$. Angluin-Eisenstat observed the bound $\gtrsim n^{-3/2}$ when $k=2$. We prove there is a universal $c>0$ such that, for all $k\ge 2$, there exists a $c_k>0$ and k integers in $\{1,2,\dots ,n\}$ with$0<\Vert \sqrt{a_1}+\dots +\sqrt{a_k}\Vert \le c_k\cdot n^{-c\cdot k^{1/3}},$ where $\Vert \cdot \Vert$ 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.