Do I understand the concept of $x^{0.84}$ correctly? I'm trying to understand the concept of $x^{0.84}$ that Jeffrey Lagarias found for Collatz Conjecture. If I'm wrong, please correct me with an answer. I understand such that,
Suppose the interval $\left[1, 2^{1\,000\,000} \right]$ is given. In this interval we have at least $\lfloor{(2^{1\,000\,000}})^{0.84}\rfloor=2^{840\,000}$ natural numbers which is goes to $1$. In other words, we can choose at least such $2^{840\,000}$ natural numbers from this sequence $\left\{1,2,3,4,5,6,7,8,9\cdots 2^{1\,000\,000} \right\}$, which goes to $1$. Or, there are  at least $2^{840\,000}$ natural numbers  in this interval $\left[1, 2^{1\,000\,000} \right]$ (But, we don't know exactly what the numbers are ?), which gives result $1$. So, Collatz Conjecture is correct for at least $2^{840\,000}$ natural numbers.  Is my understanding correct?

$x^{0.84} = x^\frac{21}{25} = (x^{21})^\frac{1}{25}$, which is the unique solution $y$ of $y^{25} = x^{21}$. For irrational $y$, $x^y = \sup_{z\in\mathbb{Q}\cap(-\infty,y)}x^z = \inf_{z\in\mathbb{Q}\cap(y,\infty)} x^z$.
