A direct sum decomposition of the integers and a question of Y. Ito
type: Item
instance of: scholarly article
title: A direct sum decomposition of the integers and a question of Y. Ito
zbMATH Open document ID: 1037.11004
published in: Tokyo Journal of Mathematics
publication date: 2004-03-30T00:00:00Z
review text: This paper is concerned with the direct sum \(Z=A\oplus C\) where \(A=\{\sum_{i\geq 0}\varepsilon_i 2^{2i+1}:\varepsilon_i\in\{0,1\}\) and \(\varepsilon_i=1\) for finitely many \(i\)'s\} and the sum is understood to be unique, namely, if \(a+c=a'+c'\) then \(a=a'\) and \(c=c'\). Let \(C\) be a subset of \(\mathbb Z\) containing \(0\). If \(C\) belongs to the family of all complements of \(A\), then (i) For every \(c,c'\in C\) either \(c=c'\) or the maximal number \(i\) such that \(2^i\) divides \(c-c'\) is even, (ii) \(C\) is maximal with respect to (i). That is if \(C'\) satisfies (i) and \(C\subset C'\) then \(C=C'\). \textit{Y. Ito} [Tokyo J. Math. 18, 259--270 (1995; Zbl 0872.11008)] proved the following: Let \(C\) be a subset of \(\mathbb Z\) containing \(0\). Then \(C\) belongs to the family of all complements of \(A\) if and only if (i), (ii) and (iii) There exists an \(\omega\) such that \(A\oplus C\supset B_\omega\), where \(B_\omega\) is a family of complements of \(A\) defined by \(B\omega=\{\sum_{i\geq 0}\varepsilon_i\omega_i 2^{2i}:\varepsilon_i\in\{0,1\}\) and \(\varepsilon_i=1\) for finitely many \(i\)'s\} with \(\omega\in\{-1,1\}^\mathbb N\) and \(\omega_i=-1\) for infinitely many \(i\)'s. He also conjectured that the third condition could be replaced with (vi) There exists an element \(D\) of the family of all complements of \(A\) such that \(A\oplus C\supset D\). In this paper a counterexample to this conjecture is presented. Please also refer to the works by \textit{R. J. Tijdeman} [Lond. Math. Soc. Lect. Note Ser. 215, 261--276 (1995; Zbl 0824.11006)] and by \textit{M. Dateyama} and \textit{T. Kamae} [Tokyo J. Math. 21, 433--440 (1998; Zbl 0921.11008)].
Mathematics Subject Classification ID: 11B13
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 2063684
zbMATH Keywords: direct sum decompositions
zbMATH Keywords: Radix representation
zbMATH Keywords: additive bases
author: Stanley Eigen
reviewed by: Takao Komatsu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1244208604
OpenAlex ID: W2062281948
cites work: Q5797130
cites work: Tiling the integers with translates of one finite set
cites work: Q4017456
cites work: On direct sum decomposition of integers and Y. Ito's conjecture
cites work: Direct sum decomposition of the integers
cites work: Addition theorems for sets of integers
cites work: Direct sum subset decompositions of Z
cites work: Rédei-matrices and applications
DOI: 10.3836/TJM/1244208604
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description: scientific article; zbMATH DE number 2063684