Direct sum decomposition of the integers
Denote by \(\mathbb{Z}(\mathbb{N})\) the set of all integers (all positive integers). If \(A,B\subseteq\mathbb{Z}\), then we put \(A+B= \{a+b: a\in A, b\in B\}\), \(A-B= \{a-b:a\in A, b\in B\}\). If \(A+B=\mathbb{Z}\) and every \(z\in\mathbb{Z}\) can be uniquely expressed in the form \(z=a+b\), then we write \(A\oplus B=\mathbb{Z}\) and \(\mathbb{Z}\) is said to be a direct sum of sets \(A\), \(B\). Similar notions can be introduced also for decomposition of the set \(\mathbb{N}\). While it is not difficult to characterize the infinite subsets that appear as summands in \(\mathbb{N}= A\oplus B\), the situation is more complicated for the case of decomposition of the set \(\mathbb{Z}\). The main result of the paper gives a characterization of sets \(C\) such that \(A\oplus C=\mathbb{Z}\), where \[ A=\Biggl\{\sum_{i\geq 0}\alpha_i2^{2i+1}: \alpha_i=1\text{ for a finite number of }i\text{'s and }\alpha_i= 0\text{ for other }i\text{'s}\Biggr\}.\tag{1} \] If \(n\in\mathbb{Z}\), \(n\neq 0\), then \(\text{ord}_2(n)\) denotes the number \(k\geq 0\) such that \(2^k\mid n\) and \(2^{k+1}\nmid n\). Further denote by \(\Omega^-\) the set of all sequences \(\omega=(\omega_1,\omega_2,\dots,\omega_n,\dots)\) such that \(\omega_n=-1\) for infinitely many \(n\)'s and \(\omega_n=1\) for other \(n\)'s. The main result: Let \(A\) be defined as in (1). Then \(A\oplus C=\mathbb{Z}\) holds if and only if \(C\) satisfies the following conditions: (i) For each \(n\in C-C\), \(n\neq 0\), the number \(\text{ord}_2(n)\) is even. (ii) \(C\) is maximal with respect to property (i). (iii) \(A\oplus C\supset B_\omega\), where \[ B_\omega= \Biggl\{\sum_{i\geq 0}\beta_i\omega_i2^{2i}: \beta_i=1\text{ for a finite number of }i\text{'s and }\beta_i=0\text{ for other }i\text{'s}\Biggr\}. \] At the end a conjecture related to the main result is formulated. Many illustrative examples are given.
scientific article; zbMATH DE number 879997