On the distribution of the values of functions on the Fibonacci sequence. (Q1432675)

Let \(F_n\) denote the Fibonacci numbers (\(F_0=1, F_1=2\) and \(F_{n+1}=F_n+F_{n-1}\)), \[ S_m(h;a)=\sum_{n=0}^{h-1}\exp\left(2\pi i\frac{aF_n}m\right), \] and \(N_m(\lambda)= \{0\leq a\leq m-1\colon |S_m(h;a)|<\sqrt{\lambda h}\}\), where \(m\) is a natural number, and \(\lambda>0\). The authors prove (Theorem 1) that if \(m\to\infty\), \(h(m)\to\infty\) and \(h(m)\leq(1/2)\log_\alpha m\), where \(\alpha=(\sqrt{5}+1)/2\), then \(\lim_{m\to\infty}N_m(\lambda)/m=1-e^{-\lambda}\). The continuous analogue of this result is stated in Theorem 2. The central limit theorem for uniformly distributed values of fractional parts of exponentially increasing sequences is formulated in Theorem 4.