In the end of the Abstract of the paper Minsky and Papert - Unrecognizable Sets of Numbers, 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".

In the end of the Abstract of the paper Minsky and Papert - Unrecognizable Sets of Numbers, 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".

In the end of the Abstract of the paper [Unrecognizable Sets of Numbers] (https://dl.acm.org/doi/pdf/10.1145/321328.321337), the authors write ".....for every infinite regular set $A$ there is a nonregular set $A'$ for which $$ |\pi_A(n)-\pi_A'(n)|\leq 1", $$ 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 $$ |\pi_B(n)-\pi_B'(n)|\leq 1\quad ? $$ If I understand right the "regular set" in this paper means "automatic set". Thanks for your help!!

In the end of the Abstract of the paper Minsky and Papert - Unrecognizable Sets of Numbers, 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".