Additive problems in positive integers with binary expansions of a special type (Q2857818)

The author of this paper study the presentation of natural numbers in the binary number system. Let \(\mathbb{N}_{0}\) be the set of positive integer numbers having even number of binary digits 1 and \(\mathbb{N}_{1}=\mathbb{N} \backslash \mathbb{N}_{0}\). He proves two theorems:NEWLINENEWLINETheorem 1. Let \(h = 0,1\) or 2. Let \(I_{j,k}(X,h)\) be the number of solutions of the equation \(n-3m=h\) in positive integers \(n\leq X\), \(n\in\mathbb{N}_{j}\), \(m\in\mathbb{N}_{k}\), where \(j\) and \(k\) are equal to 0 or 1. Then \(I_{j,k}(X,h) = \frac{X}{4} + O(X^\lambda)\), \(\lambda = \frac{\ln3}{\ln4} = 0.79\ldots\).NEWLINENEWLINETheorem 2. Let \(l = 0, 1, 2, 3\) or 4. Let \(I_{j,k}(X)\) be the number of solutions of the equation \(n-5m = l\) in positive integers \(n\leq X\), \(n\in\mathbb{N}_{j}\), \(m\in\mathbb{N}_{k}\), where \(j\) and \(k\) are equal to 0 or 1. Then \(I_{j,k}(X,h) = \frac{X}{4} + O(X^\lambda)\), \(\lambda = \frac{\ln3}{\ln4} = 0.79\ldots\).