On the distribution of the sequence of fractional parts of a slowly increasing exponential function (Q1966218)

The following is proved. Suppose that \(\gamma>0\) is a fixed real number, \(N\) is a positive integer variable, \(q=e^{\gamma/N}\), and \(\alpha= \alpha(N)\), \(\tau= \tau(N)\), and \(l=l(N)\) are real functions of \(N\) such that \(0<l\leq \tau N\leq \gamma^2 e^\gamma \alpha<N\) for all \(N\) and \(l\gg \sqrt{\alpha\ln N}\) as \(N\to \infty\). For any subset \(\Delta\subset [0,1)\) let \(Q_\Delta(N)\) denote the number of positive integers \(n\leq N\) for which \(\{\alpha q^n\}\in \Delta\). For \[ \Delta_\tau(\lambda)= \begin{cases} [\lambda,\lambda+\tau) &\text{if }0\leq\lambda\leq 1-\tau\\ [\lambda,1)\cup [0,\lambda+\tau-1) &\text{if }1-\tau< \lambda< 1\end{cases} \] the following estimate holds as \(N\to\infty\): \[ \text{mes} \{\lambda\in [0,1): |Q_{\Delta_\tau(\lambda)}(N)- \tau N|>l\}= O\biggl( \frac{\alpha\ln N}{l^2}\biggr). \] The most interesting particular case of this theorem is if \(\gamma\) and \(\tau\) are fixed real positive numbers with \(1< \tau< \gamma^2 e^\gamma<1\), \(N\) is a natural variable, \(\alpha=N\), \(q= e^{\gamma/N}\), \(\Psi(N)\) is a real function of \(N\) with \(0< \Psi(N)\leq \tau\sqrt{N/\ln N}\) and \(\Psi(N)\to \infty\) as \(N\to \infty\), then \[ \text{mes} \{\lambda\in [0,1): |Q_{\Delta_\tau(\lambda)}(N)- \tau N|> \sqrt{N\ln N}\cdot \Psi(N)\}= O\biggl( \frac{1}{\Psi^2(N)} \biggr). \]