Given a real number $r$, and an integer $b$>0, we can define $B_b(r)$ as the set of numbers which are obtained from $r$ by writing $r$ in base $b$ and then altering a density zero subset of its digits. For example, if $r=0.0000...$ then $B_2(r)$ would include $0.10100100001000001 \cdots$ . There is a slight ambiguity here in the definition if r has a finite expansion in base $b$; for example whether for 1 we use $r$ written as 1.0000... or we use r written as 0.9999... For our purposes, it is probably ok to include both, and won't alter the answers to any of the things we care about. Given S a subset of the real numbers we can then define $B_b(S)$ as the union of $B_b(s)$ for every s in S. We'll define $T(S)$ as the union of $B_b(S)$ for every base $b$.

Set $U(r)$ to be $\bigcup_{i=0}^{\infty} T^i(r)$.

Question 1) Is it true that $U(0)= \mathbb{R}$?. My guess is that the answer is probably "No" but it isn't obvious. One might try a measure theoretic argument, but since the union of an uncountable number of measure zero sets is not necessarily of measure zero, it doesn't seem to get a result. Note that for any $r$, $T(r)$ contains every number of the form $r+q$ for $q$ a non-negative rational, since adding $\frac{a}{b}$ changes only a finite number of digits in base $b$.

We can also define versions of $B_b$ for complex numbers, where we do the same thing changing a zero density set of the real part and the complex part, and then define $T_{\mathrm{com}}(s)$ accordingly. Given a set of complex numbers Let $\overline{S}$ be the algebraic closure of the smallest field containing a given set S of complex numbers. We can then define $U_{\mathrm{com}}(r)$ as the union of $T(r)$, $\overline{(T(r))}$, $T(\overline{(T(r))})$, $\overline{T(\overline{(T(r))})} \cdots$

Question 2: Is $U_{\mathrm{com}}(0)$ all complex numbers? This question seems to be possibly substantially much harder, and I'm less certain what the answer is.