On monochromatic sums of squares and primes
type: Item
instance of: scholarly article
title: On monochromatic sums of squares and primes
zbMATH Open document ID: 1120.11005
author: Norbert Hegyvári
author: François Hennecart
published in: Journal of Number Theory
publication date: 2007-05-10T00:00:00Z
review text: For a subset \({\mathcal A}\subseteq {\mathcal U}\), one typical inverse additive question is whether any element in \(\mathcal U\) is a sum of elements in \(\mathcal A\), and \(\mathcal A\) is called an additive basis of \(\mathcal U\) if so. \(\mathcal A\) is called the additive basis of order \(h\) if any element of \(\mathcal U\) is a sum of at most \(h\) elements of \(\mathcal A\). In the case of \({\mathcal U}={\mathbb Z}\), \(\mathcal A\) is called an asymptotic basis if any large enough integer is a sum of elements in \(\mathcal A\). The asymptotic order can be defined in a similar way. Another variation is to add the coloring to the problem. A coloring of \(\mathcal A\) is a partition of \(\mathcal A\). A monochromatic sum is a sum taken from one of the sets in the given partition. For any positive integer \(K\), we can define the \(K\)-chromatic order ord\(_K\) as the maximum order of monochromatic sum over any \(K\)-partition. This paper looks at two special sets: \({\mathcal A}={\mathcal Q}\), the set of squares, and \({\mathcal A}={\mathcal P}\), the set of primes. For \({\mathcal A}={\mathcal Q}\), the paper gives the following upper and lower bounds \[ K\exp\left((\log2+o(1))\frac{\log K}{\log\log K}\right)\leq\text{ ord}_K(\mathcal Q)\leq c_3(K\log K)^5 \] where \(c_3\) can be taken equal to \(10^9\). For \({\mathcal A}={\mathcal P}\), the following upper and lower bounds are given \[ (e^\gamma+o(1))K\log\log K\leq\text{ ord}_K(\mathcal Q)\leq 1500K^3. \] In the case of a hybrid scenario, \({\mathcal A}=\{1\}\cup\{hq\,| \,q\geq0\}\), not every basis has a \(K\)-chromatic order, which leads to the definition of \textit{well sifted}. For any prime \(p\), denote \({\mathcal A}^{(p)}=\{a\in{\mathcal A}\,| \,p| a\,\}\). For any real number \(\tau<1\), \(\mathcal A\) is well \(\tau\)-sifted if there exists a real number \(\tau>0\) such that \[ \mathop{\lim\sup}_{N\to\infty}\frac{A^{(q_1,q_2,\dots,q_K)}(N)}{A(N)}\leq1-\tau \] For a well \(\tau\)-sifted basi…
reviewed by: Chuang Peng
Mathematics Subject Classification ID: 11B13
Mathematics Subject Classification ID: 11P70
Mathematics Subject Classification ID: 11B05
zbMATH DE Number: 5151586
zbMATH Keywords: monochromatic sum
zbMATH Keywords: square
zbMATH Keywords: prime
MaRDI profile type: Publication
full work available at URL: j.jnt.2006.09.007
OpenAlex ID: W2053903242
cites work: Q3693570
cites work: Q4889853
cites work: Additive properties of dense subsets of sifted sequences
cites work: Finite addition theorems. II
cites work: Q4256474
cites work: Unsolved problems in number theory
DOI: 10.1016/J.JNT.2006.09.007
description: scientific article; zbMATH DE number 5151586