The sequence of fractional parts of roots

TL;DR

This paper investigates the behavior of the fractional parts of roots, extending previous results to a broader set of real numbers and demonstrating that most t have predictable asymptotic properties with only finitely many exceptions.

Contribution

The paper generalizes prior work by removing the rationality condition on log(t) and shows that almost all t have only finitely many exceptions in the sequence of fractional parts of roots.

Findings

Most t have asymptotic behavior with finitely many exceptions.

Uncountably many t have only finitely many exceptional n.

Explicit examples of t with infinitely many exceptions are provided.

Abstract

We study the function M(t,n) = Floor[ 1 / {t^(1/n)} ], where t is a positive real number, Floor[.] and {.} are the floor and fractional part functions, respectively. In a recent article in the Monthly, Nathanson proved that if log(t) is rational, then for all but finitely many positive integers n one has M(t,n) = Floor[ n / log(t) - 1/2 ]. We extend this by showing that, without condition on t, all but a zero-density set of integers n satisfy M(t,n) = Floor[ n / log(t) - 1/2 ]. Using a metric result of Schmidt, we show that almost all t have asymptotically log(t) log(x)/12 exceptional n<x. Using continued fractions, we produce uncountably many t that have only finitely many exceptional n, and also give uncountably many explicit t that have infinitely many exceptional n.