Sums of square roots that are close to an integer

2024/01/18 by Steinerberger, Stefan
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2401.10152

Abstract

Let k ∈ ℕ and suppose we are given k integers 1 ≤ a₁, …, a_k ≤ n. If √(a₁) + … + √(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 ≥ 2, there exists a c_k > 0 and k integers in \1,2,…, n\ with 0 lt;‖√(a₁) + … + √(a_k) ‖ ≤ c_k⋅ n^-c ⋅ k^1/3, 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: even for k=3, constructing explicit examples of integers whose square root sum is nearly an integer appears to be nontrivial.

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