Is there a standard notation for this set of numbers $$\left\{\frac{K}{3^n} : K\in\mathbb N,\  n\in \mathbb N \right\}$$
Any positive integer divided by any power of $3$.
 A: First, let me say a bit about the same set, but with "$\mathbb{N}$" replaced with "$\mathbb{Z}$" (to allow negatives). I would call these the tryadic rationals in analogy with the dyadic rationals.
In terms of notation, they are a direct limit, and can be written as $$\lim_{\rightarrow}3^{-i}\mathbb{Z}.$$ I think the notation "$3^{-\infty}\mathbb{Z}$" would probably be understandable, in a somewhat strained analogy with the notation for the Prufer groups $\mathbb{Z}(p^\infty)$.

Now, for the nonnegative elements, I would call them the "nonnegative tryadic rationals" - granted, that's not very snappy, but not everything needs to be. In terms of notation, I think replacing $\mathbb{Z}$ with $\mathbb{Z}$ would be good: e.g. "$3^{-\infty}\mathbb{N}$" or "$\lim_\rightarrow 3^{-i}\mathbb{N}$."
A: The members of $\displaystyle\left\{\frac{K}{3^n} : K\in\mathbb Z,\  n\in \mathbb N \right\}$ are sometimes called "ternary rational numbers" or "ternary rationals", but I don't know of a standard notation for them besides using that phrase or writing the expression that appears here.  With $K\in\mathbb N$ rather then $K\in\mathbb Z$, I'd call the members "positive ternary rationals".
