Sums of square roots that are close to an integer
type: Item
instance of: scholarly article
title: Sums of square roots that are close to an integer
author: Stefan Steinerberger
published in: Journal of Number Theory
publication date: 2024-06-17T00:00:00Z
review text: In this article, the author examines the following question: Given \(k,n \in \mathbb{N}\), and suppose that integers \(1 \leq a_1,\ldots,a_k \leq n\) are given with the property that \(\sum_{i = 1}^k \sqrt{a_i} \notin \mathbb{N}\), how close to an integer could such a sum come? While surveying results on similar-looking problems such as minimizing the distances\N\[\N\left \vert\sum_{i = 1}^k \sqrt{a_i} - \sum_{i = 1}^k \sqrt{b_i} \right\rvert,\N\]\Nwhich appears in numerical analysis, or considering the gap and fine-scale distributions of \(\sqrt{n} \pmod 1\) that has a rich body of literature in dynamics and number theory, the author proves the following main result:\N\NThere exists \(c > 0\) such that for any \(k \in \mathbb{N}\) fixed, there exists \(c_k > 0\) such that for all \(n \in \mathbb{N}\), there exist \(1 \leq a_1, \ldots, a_k \leq n\) with \N\[\N0 < \lVert \sqrt{a_1} + \ldots + \sqrt{a_k} \rVert \leq c_k n^{-c \cdot k^{1/3}}.\tag{1}\N\]\NActually, the author proves the stronger result that the largest gap arising from the set \(\left\{\sqrt{a_1} + \ldots + \sqrt{a_k} \pmod 1: 1 \leq a_1, \ldots, a_k \leq n\right\}\) is bounded by the quantity described above. The proof of (1) uses standard tools from the theory of exponential sums (i.e. van der Corput and Vinogradov method).\N\NAs to be expected, the bottleneck for improvements in the author's approach is getting a better control on the exponential sums. While some vaguely formulated form of exponent pair hypothesis could provide \(n^{-k^{1/2 + \varepsilon}}\) and modeling of \(\sqrt{a_i} \pmod 1\) with uniformly distributed random variables suggests a bound to be of the form \(\approx n^{-k}\), such a result would still be far away from matching the lower bound: By an elementary approach presented in the article, the best lower bound known is of the form \(\gg n^{1/2 - 2^{k-1}}\), and the author claims that a polynomial bound in \(k\) seems out of reach.
reviewed by: Manuel Hauke
Mathematics Subject Classification ID: 11J71
Mathematics Subject Classification ID: 37A44
Mathematics Subject Classification ID: 65D18
Mathematics Subject Classification ID: 68Q17
zbMATH DE Number: 7865938
zbMATH Keywords: square root sum problem
zbMATH Keywords: \(\sqrt{n} \bmod 1\)
zbMATH Keywords: sum of square roots
zbMATH Keywords: exponential sums
MaRDI profile type: Publication
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DOI: 10.1016/J.JNT.2024.03.002
description: scientific article; zbMATH DE number 7865938