  Partitioning a set into similar, if not, identical, parts is a fundamental
research topic in combinatorics. The question of partitioning the integers in
various ways has been considered throughout history. Given a set $\{x_1,
\ldots, x_n\}$ of integers where $x_1<\cdots<x_n$, let the {\it gap sequence}
of this set be the nondecreasing sequence $d_1, \ldots, d_{n-1}$ where $\{d_1,
\ldots, d_{n-1}\}$ equals $\{x_{i+1}-x_i:i\in\{1,\ldots, n-1\}\}$ as a
multiset. This paper addresses the following question, which was explicitly
asked by Nakamigawa: can the set of integers be partitioned into sets with the
same gap sequence? The question is known to be true for any set where the gap
sequence has length at most two. This paper provides evidence that the question
is true when the gap sequence has length three. Namely, we prove that given
positive integers $p$ and $q$, there is a positive integer $r_0$ such that for
all $r\geq r_0$, the set of integers can be partitioned into $4$-sets with gap
sequence $p, q$, $r$.
