  The Three Gap Theorem, also known as the Steinhaus Conjecture, is a classical
result on the combinatorics of the fractional part function, and has since been
generalized in many ways. In this paper, we pose a new problem related to these
results: for which other periodic functions does an analogue of the Three Gap
Theorem hold? We prove analogous results for certain classes of
piecewise-linear periodic functions and demonstrate the existence of functions
for which no bound exists on the number of gap lengths.
