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The sequence of fractional parts of roots

Volume 169 / 2015

Kevin O'Bryant Acta Arithmetica 169 (2015), 357-371 MSC: Primary 11B83, 11J99, 11J70. DOI: 10.4064/aa169-4-4

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 $n$.

Authors

  • Kevin O'BryantDepartment 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.
    e-mail

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