The sequence of fractional parts of roots
Volume 169 / 2015
Abstract
We study the function $\def\fp#1{\{#1\}}\def\tfloor#1{\lfloor #1 \rfloor}M_\theta(n)=\tfloor{1/\fp{\theta^{1/n}}}$, where $\theta$ is a positive real number, $\def\tfloor#1{\lfloor #1 \rfloor}\tfloor{\cdot}$ and $\def\fp#1{\{#1\}}\fp{\cdot}$ are the floor and fractional part functions, respectively. Nathanson proved, among other properties of $M_\theta$, that if $\log\theta$ is rational, then for all but finitely many positive integers $n$, $\def\tfloor#1{\lfloor #1 \rfloor}M_\theta(n)=\tfloor{n/\!\log\theta-1/2}$. We extend this by showing that, without any condition on $\theta$, all but a zero-density set of integers $n$ satisfy $\def\tfloor#1{\lfloor #1 \rfloor}M_\theta(n)=\tfloor{n/\!\log\theta-1/2}$. Using a metric result of Schmidt, we show that almost all $\theta$ have asymptotically $(\log\theta \log x)/12$ exceptional $n \leq x$. Using continued fractions, we produce uncountably many $\theta$ that have only finitely many exceptional $n$, and also give uncountably many explicit $\theta$ that have infinitely many exceptional $n$.
