On the number of solutions of some Diophantine equations in positive integers with given properties of binary expansions (Q2857815)

Let \(N_0\) be the set of natural numbers that contain an even number of 1's in their binary representations and \(N_1 = \mathbb N \backslash N_0\). Let \(s, j \in \{0, 1 \}\) and NEWLINE\[CARRIAGE_RETURNNEWLINEJ(X, h, k) = \sum_{_{\substack{ m-kn=h \\ n \leq X \\ n \in N_s; \;kn+h \in N_j }}} 1.CARRIAGE_RETURNNEWLINE\]NEWLINENEWLINEThe following two assertions are proved.NEWLINENEWLINENEWLINETheorem 1. Let \(X, H\) be natural numbers. ThenNEWLINENEWLINE\[CARRIAGE_RETURNNEWLINE\sum_{h=0}^{H-1} \left| J(X, h, 1) - \frac{X}{4} \right| = O(XH^x) + O(H^2),CARRIAGE_RETURNNEWLINE\]NEWLINE where \(x = \log_2 (1 + \sqrt{5}) - 1 = 0.694\ldots\).NEWLINENEWLINENEWLINETheorem 2. Let \(k > 1, X, h\) be natural numbers. ThenNEWLINENEWLINE\[CARRIAGE_RETURNNEWLINEJ(X, h, k) = \frac{X}{4} + O\left ( h.m\left ( \frac{X}{h} \right ) \right ) + O((kX)^{\lambda}),CARRIAGE_RETURNNEWLINE\]NEWLINE where NEWLINE\[CARRIAGE_RETURNNEWLINEm(t) = \min \left ( \tfrac{8}{11} \sqrt{k}t^{\psi + \frac{c}{\ln k}}, \tfrac{8}{3} t^{1-\frac{c_0}{\log_2 k+3}} \right ),CARRIAGE_RETURNNEWLINE\]NEWLINE NEWLINE\[CARRIAGE_RETURNNEWLINE\psi = \frac{8 - \log_2 (\sqrt{5} + 1)}{10 - 2\log_2 (\sqrt{5} + 1)} = 0.9537 ...,CARRIAGE_RETURNNEWLINE\]NEWLINE NEWLINE\[CARRIAGE_RETURNNEWLINEc = \frac{\ln 33}{5 - \log_2 (\sqrt{5} + 1)} = 1.057\ldots,\quad c_0 = 2 - \log_2 3 = 0.4150\ldots,\quad \lambda = \frac{\ln 3}{\ln 4} = 0.7924\ldots.CARRIAGE_RETURNNEWLINE\]