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Sums of square roots that are close to an integer

Stefan Steinerberger

Abstract

Let $k \in \mathbb{N}$ and suppose we are given $k$ integers $1 \leq a_1, \dots, a_k \leq n$. 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 $k \geq 2$, there exists a $c_k > 0$ and $k$ integers in $\left\{1,2,\dots, n\right\}$ with $$ 0 <\|\sqrt{a_1} + \dots + \sqrt{a_k} \| \leq c_k\cdot n^{-c \cdot k^{1/3}},$$ where $\| \cdot \|$ 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.

Sums of square roots that are close to an integer

Abstract

Let and suppose we are given integers . If is not an integer, how close can it be to one? When , the distance to the nearest integer is . Angluin-Eisenstat observed the bound when . We prove there is a universal such that, for all , there exists a and integers in with 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 , constructing explicit examples of integers whose square root sum is nearly an integer appears to be nontrivial.
Paper Structure (9 sections, 4 theorems, 52 equations)

This paper contains 9 sections, 4 theorems, 52 equations.

Table of Contents

  1. Introduction and Results
  2. Introduction
  3. The sum of square roots problem
  4. The case $k = 3$.
  5. Main Result.
  6. Exponential sum estimates
  7. Proof
  8. Proof of the Theorem.
  9. Concluding Remarks.

Key Result

Theorem 1

Let $1 \leq a,b \leq n$. If $\sqrt{a} + \sqrt{b} \notin \mathbb{N}$, then

Theorems & Definitions (5)

  • Theorem : Angluin & Eisenstat angluin
  • Proposition 1
  • proof
  • Theorem
  • Theorem : Vinogradov iwaniec