Atypical Numbers and Diophantine Approximation

For any positive real number $\theta > 1$, the sequence $\theta^{1/n}$ goes to 1. $\def\fp{\mathop{\rm fp}\nolimits}$

Nathanson and O'Bryant wrote papers studying the details of this convergence and discovered some truly amazing properties. Let $\fp$ denote "fractional part." One critical discovery is that, for almost all $n$, $\lfloor{\frac{1}{\fp{\theta^{1/n}}}}\rfloor$ is equal to $\lfloor{n/\log\theta-1/2}\rfloor$, the exceptions being termed "atypical" $n$, and that for $\log\theta$ rational, the number of atypical $n$ is finite. Nathanson left, as an open question, among others, whether $\theta$'s with irrational logs can have infinite or only finite atypical $n$. O'Bryant developed a theory to answer these questions and constructed infinite families of bounded $\theta$'s, some with no atypical $n$, and some with infinite atypical $n$. However, he left as an open problem whether there are uncountably many infinite families of $\theta$ with irrational logs that are not bounded. This paper shows the restriction of boundedness cannot be removed.

David Seff