As a straightforward generalization of IMO Shortlist 2001 problem C4, we can show the following fact:
Let $u$ and $v$ be two positive integers. A set of three integers $\left\lbrace x,y,z\right\rbrace $ with $x < y < z$ is called historic if $\left\lbrace z-y,y-x\right\rbrace =\left\lbrace u,v\right\rbrace$. Then, the set of all nonnegative integers can be written as a union of pairwise disjoint historic sets.
This is considered a medium-difficulty exercise, and the solutions avaliable online are basically one solution in different writeups (ok, 3 and 4 aren't even different).
As far as I understand, this solution doesn't answer any of the following
Questions:
(1) Is there a periodic way to write the set of all nonnegative integers as a union of pairwise disjoint historic sets? Equivalently, is there a function $f:\left\lbrace 0,1,2,...\right\rbrace \rightarrow\mathbb{N}$ such that $\left\lbrace x\in\left\lbrace 0,1,2,...\right\rbrace \mid f\left( x\right) =d\right\rbrace $ is a historic set for every $d\in\mathbb{N}$, and such that $f\left(x+u\right)=f\left(x\right)+v$ for some fixed positive integers $u$ and $v$ and all sufficiently large $x$ ? Actually, I believe that the answer is yes here, as can be deduced from the above problem using an additional argument.
(2) Is there a purely periodic way to do this? Equivalently, is there a function $f:\left\lbrace 0,1,2,...\right\rbrace \rightarrow\mathbb{N}$ such that $\left\lbrace x\in\left\lbrace 0,1,2,...\right\rbrace \mid f\left( x\right) =d\right\rbrace $ is a historic set for every $d\in\mathbb{N}$, and such that $f\left(x+u\right)=f\left(x\right)+v$ for some fixed positive integers $u$ and $v$ and all $x$?
(3) Is there a positive integer $n$ such that the set $\left\lbrace 0,1,2,...,n\right\rbrace $ is a union of pairwise disjoint historic sets?
(4) What happens if we start with three positive integers $u$, $v$, $w$, and call a set of four integers $\left\lbrace x,y,z,t\right\rbrace $ with $x < y < z < t$ historic if $\left\lbrace t-z,z-y,y-x\right\rbrace =\left\lbrace u,v,w\right\rbrace $ ?
I would not be surprised to see very easy counterexamples to some of these questions, yet I have not been able to find one myself. The same holds for proofs (no progress since 2008).
