A direct sum decomposition of the integers and a question of Y. Ito
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)].
scientific article; zbMATH DE number 2063684