TITLE: What restrictions on decimal expansions lead to countably infinite subsets of the reals?
QUESTION [5 upvotes]: Consider real numbers $S : x \in [0,1]$ whose decimal expansions are $x = 0.d_1 d_2 d_3 \ldots$.
Now institute various exclusions, listed below. I am interested to learn of 
general principles that will allow me to conclude that $S_X$ is either countable 
or uncountable.

$S_{!5}$: The decimal representation excludes all $5$'s.
$S_{5^{\textrm{th}}}$: In the decimal representation, every $5^{\textrm{th}}$ digit is $5$.
$S_{\textrm{odd}}$: The decimal representation excludes all even digits: $0,2,4,6,8$.
$S_{01}$: The decimal representation excludes all but the two digits: $0$ and $1$. 
$S_{1}$: The decimal representation uses (after the $0.$) only the digit $1$: $0.1, 0.11, 0.111, \ldots$.
$S_{\ge}$: The decimal representation is non-decreasing: successive digits are the same or larger. E.g., $.1144456777777788999\ldots$.
$S_{k\pm}$: The decimal representation $k$-oscillates: 
The sequence consists of $k$ or more digits (non-strictly) increasing, followed by a sequence of $k$ or more digits that (non-strictly) decreases, and so on. E.g., for $k=5$, $0.11339 \; 966432 \; 567777 \; \ldots$.
$S_{\textrm{max/min}}$: The decimal representation is finitely oscillatory: there are only a finite number of digit minima and maxima in the sequence of digits.

Perhaps each case must be handled separately?
I am particularly interested in difficult, borderline cases that are not so 
straightforward to settle, which could be used as good student exercises
to distinguish countable from uncountable.

REPLY [3 votes]: Don't know that I'd call it a general principle, but most of these follow from remapping the restricted set to the reals, perhaps in another basis.


$S_{!5}$: The decimal representation excludes all $5$'s.


Decrement all digits larger than $5$, then the set becomes the representation of the entire $[0,1]$ in base $9$, thus uncountable.


$S_{5^{\textrm{th}}}$: In the decimal representation, every $5^{\textrm{th}}$ digit is $5$.


Drop every $5^{th}$ digit, then the set becomes the decimal representation of the entire $[0,1]$, thus uncountable.


$S_{\textrm{odd}}$: The decimal representation excludes all even digits: $0,2,4,6,8$.


Remap the $5$ remaining digits $1,3,5,7,9$ to $0,1,2,3,4$, then the set becomes the representation of the entire $[0,1]$ in base $5$, thus uncountable.


$S_{01}$: The decimal representation excludes all but the two digits: $0$ and $1$.


That gives the representation of the entire $[0,1]$ in base $2$, thus uncountable.


$S_{1}$: The decimal representation uses (after the $0.$) only the digit $1$: $0.1, 0.11, 0.111, \ldots$.


This can be indexed by the number of decimal digits, thus countable.


$S_{\ge}$: The decimal representation is non-decreasing: successive digits are the same or larger. E.g., $.1144456777777788999\ldots$.


The sequence of digits will become constant eventually, thus countable.


$S_{k\pm}$: The decimal representation $k$-oscillates: 
  The sequence consists of $k$ or more digits (non-strictly) increasing, followed by a sequence of $k$ or more digits that (non-strictly) decreases, and so on. E.g., for $k=5$, $0.11339 \; 966432 \; 567777 \; \ldots$.


Drop every other group of $k$ digits, and truncate the remaining ones to the first $k$ digits, which leaves groups of exactly $k$ increasing decimal digits. Consider each such group as one digit in some base made up of all increasing sequences of $k$ decimal digits. Then the set becomes the representation of the entire $[0,1]$ in that base, thus uncountable.


$S_{\textrm{max/min}}$: The decimal representation is finitely oscillatory: there are only a finite number of digit minima and maxima in the sequence of digits.


The sequence of digits will become constant eventually, thus countable.

[ EDIT ]   Numbered the points to stay in sync with the OP, and made a minor change to #7 which I had first (mis)read as exactly $k$ digits.