# Does writing the infinite binary strings with the radix two and one, still give an exact cover of the two-adic numbers?

Question

Does writing the infinite binary strings with the radix two and one, still give an exact cover of the two-adic numbers?

Unless I'm mistaken, $$\displaystyle \Bbb Z_2=\left\{\sum_{i=0}^\infty2^ia_i:a_i\in\{0,1\}\right\}$$ is an exact cover of the 2-adic integers.

And we can substitute any 2-adic unit for one in the radix, so for example:

$$\displaystyle \Bbb Z_2=\left\{\sum_{i=0}^\infty2^ia_i:a_i\in\{0,5\}\right\}$$

But what about substituting for zero? Can we substitute any number equivalent to $$0\pmod 2?$$ It's not so clear to me whether it is also the case that $$\displaystyle \Bbb Z_2=\left\{\sum_{i=0}^\infty2^ia_i:a_i\in\{2,1\}\right\}$$? I don't see how that would work.

## 1 Answer

For any prime element $$\pi \in \mathbb Z_p$$ and every fixed set of representatives $$R \subset \mathbb Z_p$$ of $$\mathbb Z_p/p\mathbb Z_p \simeq \mathbb Z_p/\pi \mathbb Z_p$$, each element $$x \in \mathbb Z_p$$ has a unique representation $$x = \sum_{i=0}^\infty a_i \pi^i$$ with all $$a_i \in R$$.

So instead of $$2$$ in your series expression you can take e.g. $$6$$, or $$-18$$, or $$2\sqrt{17}$$ (for either choice of $$\sqrt{17}$$ that exists in $$\mathbb Z_2$$). You cannot take "any number equivalent to $$0$$ (mod $$2$$)", namely not $$4$$ or $$-32$$ or $$100$$ or anything else that is not a prime a.k.a. uniformizer, i.e. does not have $$2$$-adic absolute value $$1$$.

Those series expansions look strange at first. In the first part of my answer to Why does this generalized ring of Witt vectors not depend on a choice of a prime element?, I wrote $$5$$-adic numbers as series expansions with the prime element $$10 \in \mathbb Z_5$$ (but keeping the popular set of representatives $$\{0,1,2,3,4\}$$ as $$R$$).

• Thank-you. This question arose because I was struggling to write $0$ using the radix $\{2,1\}$. So trying again with renewed confidence I came up with $\overline12_2=0$, correct? Commented Jun 6, 2023 at 15:16