On Hilbert cubes in certain sets
type: Item
instance of: scholarly article
title: On Hilbert cubes in certain sets
zbMATH Open document ID: 0990.11011
DOI: 10.1023/A:1009883404485
published in: The Ramanujan Journal
publication date: 2002-08-17T00:00:00Z
review text: Given \(k\geq 1\), a set \(H\subset \mathbb{N}\) is called a cube of size \(k\) if there exist \(a>0\) and \(x_1,\dots,x_k\) such that \(H=\{a+\sum_{i=1}^k\varepsilon_ix_i:\varepsilon=0\text{ or }1\}\). If the set of \(x\)'s is infinite the cube is called infinite. If \(\mathcal{A}\subset \mathbb{N}\) and \(n\in\mathbb{N}\) then \(F_{\mathcal{A}}(n)\) denotes the size of the largest Hilbert cube contained in \(\mathcal{A}\cap\{1,\dots,n\}\). Finally, let \(\mathcal{S}\), \(\mathcal{P}\), and \(\mathcal{P}_k\) denote the set of all squares, the set of all primes, and the set of positive integers composed of the primes not exceeding \(k\), respectively. The authors prove that (1) for \(n>n_0\) we have \(F_{\mathcal{S}}(n)<48(\log n)^{1/3}\), (2) for \(\varepsilon>0\), \(n>n_0(\varepsilon)\) we have \((1.1-\varepsilon)\log\log n<F_{\mathcal{P}}(n)<(16+\varepsilon)\log n\), (3) the sets \(\mathcal{P}\), \(\mathcal{P}_k\), \((k\in\mathbb{N},k\geq 2)\) do not contain an infinite cube. To prove (1) the authors are led to study the modular analogue of the problem for a fixed prime \(p\). If \(f(p)\) denotes the size of the largest cube with elements taken modulo \(p\) such that each element is a square modulo \(p\), then for \(\varepsilon>0\), \(p>p_0(\varepsilon)\) we have \((1.1-\varepsilon)\log\log p<f(p)<12 p^{1/4}\).
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 05D10
Mathematics Subject Classification ID: 11P99
Mathematics Subject Classification ID: 11N36
Mathematics Subject Classification ID: 11L03
zbMATH DE Number: 1371467
zbMATH Keywords: Ramsey-type theorems
zbMATH Keywords: Hilbert cube
zbMATH Keywords: subset sum
zbMATH Keywords: Gaussian sum
zbMATH Keywords: square
author: Norbert Hegyvári
author: András Sárközy
reviewed by: Štefan Porubský
MaRDI profile type: Publication
full work available at URL: a:1009883404485
OpenAlex ID: W2076667405
description: scientific article; zbMATH DE number 1371467