In the end of the Abstract of the paper [Minsky and Papert - Unrecognizable Sets of Numbers](https://doi.org/10.1145/321328.321337), the authors write "…for every
infinite regular set $A$ there is a nonregular set $A'$ for which
$$ \lvert\pi\_A(n)-\pi\_{A'}(n)\rvert\leq 1\text{",} $$
where $\pi\_A(n)$ is the counting function for $A$.
But I don't find a reference in the paper. Also I want to know if the following statement is true or not: "…for every
infinite nonregular set $B$ there is a regular set $B'$ for which
$$ \lvert\pi\_B(n)-\pi\_{B'}(n)\rvert\leq 1\text{."} $$
If I understand right the "regular set" in this paper means "automatic set".



A question on regular sets
The second question has a negative answer. The asymptotic behavior of $\pi\_B(n)$ and $\pi\_{B'}(n)$ would be the same, and if $\pi\_B(n)$ satisfies any of the criteria for non-regularity on page 283 of the paper by Minski and Papert, then $\pi\_{B'}(n)$ would satisfy the same, and thus, cannot be regular.


For the first question, the relevant portion of the paper is Section 5, titled "Impossibility of a Converse".


