A binary problem
Let \(n=\sum_{k=0}^{\infty} \varepsilon_k 2^k\) be the binary decomposition of the positive integer \(n\), where \(\varepsilon_k \in \{0,1\}, k=0,1,2,\dots\), and let \(\mathbb{N}_i\) denote the set of positive integers \(n\) such that \(\sum_{k=0}^{\infty} \varepsilon_k\equiv i\) (mod \(2\)), where \(i\in \{0,1\}\). Furthermore, denote by \(F_{i,j}\) the number of solutions of the equation \(n-m=1\), where \(n,m\leq x, n\in \mathbb{N}_i, m\in \mathbb{N}_j, i,j\in \{0,1\}\). The author shows that the following asymptotic formulae hold for \(i,j\in \{0,1\}\): \[ F_{i,j}=C_{i,j}x + O(\log x), \] where \(C_{0,0}= C_{1,1}=1/6\) and \(C_{1,0}= C_{0,1}=1/3\).
scientific article; zbMATH DE number 1158943