The sequence of fractional parts of roots

Kevin O'Bryant¹

¹ Department of Mathematics College of Staten Island (CUNY) Staten Island, NY 10314, U.S.A. and CUNY Graduate Center New York, NY 10016, U.S.A.

Acta Arithmetica, Tome 169 (2015) no. 4, pp. 357-371

Résumé

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 $n$.

Détail

DOI : 10.4064/aa169-4-4

Keywords: study function def def tfloor lfloor rfloor theta tfloor theta where theta positive real number def tfloor lfloor rfloor tfloor cdot def cdot floor fractional part functions respectively nathanson proved among other properties theta log theta rational finitely many positive integers def tfloor lfloor rfloor theta tfloor log theta extend showing without condition theta zero density set integers satisfy def tfloor lfloor rfloor theta tfloor log theta using metric result schmidt almost theta have asymptotically log theta log exceptional leq using continued fractions produce uncountably many theta have only finitely many exceptional uncountably many explicit theta have infinitely many exceptional nbsp