EDIT: This has now been crossposted at MO: https://mathoverflow.net/questions/219366/antirandom-reals.
This is partially motivated by my question at mathoverflow: https://mathoverflow.net/questions/218810/relative-null-ness.
For a function $f: \mathbb{N}\rightarrow\mathbb{R}_{>0}$ and a set $X\subseteq \mathbb{R}$, an $f$-cover of $X$ is a sequence $(I_n)_{n\in\mathbb{N}}$ of open intervals with rational endpoints such that
$X\subseteq\bigcup I_n$, and
$\mu(I_n)<f(n)$.
Say that a set $X\subseteq\mathbb{R}$ is $f$-small if $X$ has an $f$-cover.
Talking about $f$-covers provides us with many different refinements of the notion of "measure zero": e.g., a set is strong measure zero if it has an $f$-cover for every function $f$.
Say that a set of reals $X$ is computably strong measure zero (csmz) if there is an $e$ such that $\Phi_e^f$ is an $f$-cover of $X$ whenever $f$ is a function from $\mathbb{N}$ to $\mathbb{R}$. (This is sadly not the same as effective strong measure zero, a notion introduced by Kihara in his thesis; see also Higuchi and Kihara http://www.sciencedirect.com/science/article/pii/S016800721400044X.)
In computability theory, a real is said (informally) to be "random" if it is not in any "simple" measure-zero set; there are of course many ways to formalize this, but this is the basic theme. Csmz sets provide a very strong notion of non-randomness: say that an individual real $r$ is antirandom if $\{r\}$ is csmz - equivalently, if $r$ is contained in some csmz set. My question is:
What are the antirandom reals?
I strongly suspect that every antirandom real is computable, but I can't prove it.
TECHNICAL NOTE: the set of antirandom reals is countable - this is because there is a countable set $\{A_i: i\in\omega\}$ of csmz sets such that every csmz set is contained in one of the $A_i$s, it can be forced that every strong measure zero set is countable, and antirandomness is a $\Pi^1_1$ property (so "there are countably many antirandom reals" is absolute assuming large cardinals).
