The Prisoner's Dilemma

(with thanks to Kate Stange, University of Colorado)

Recall that an (infinite) arithmetic progression is the set of all numbers of the form $a+nd$, where $a$ and $d$ are fixed and $n$ is any natural number. For example, 5, 8, 11, 14, $\dots$, and so on forever, is an arithmetic progression of the form $5+3n$. Call a set of natural numbers “evenly spaced but essentially aperiodic” if it is the union of infinitely many arithmetic progressions, but not the union of finitely many arithmetic progressions. For example, the natural numbers larger than one are the union of all of the arithmetic progressions of the form $p+np$ for each prime number $p$, but they are not essentially aperiodic because they also form the single arithmetic progression $2+1n$.

(a) Find (with proof) an evenly spaced but essentially aperiodic set.

(b) More challenging: find two evenly spaced but essentially aperiodic sets
$A$ and $B$ such that every natural number is in exactly one of $A$ or $B$.

Of course, any answer to (b) gives you two answers to (a), so you may also want to find an answer for (a), the complement of which is not evenly spaced but essentially aperiodic.