  The probability density of a quantum particle moving freely within a circular
ring can exhibit local flow patterns inconsistent with its angular momentum, a
phenomenon known as quantum backflow. In this study, we examine a quantum
particle confined to a ring and prepared in a state composed of a fixed (yet
arbitrary) number of lowest energy eigenstates with non-negative angular
momentum. We investigate the time-dependent behavior of the probability current
at a specified point along the ring's circumference. We establish precise lower
and upper bounds for this probability current, thereby delineating the exact
scope of the quantum backflow effect. We also present an analytical expression
for a quantum state that yields a record-high backflow probability transfer,
reaching over 95% of the theoretical bound. Furthermore, our investigation
yields compelling numerical and analytical evidence supporting the conjecture
that the current-versus-time function associated with states maximizing
backflow probability transfer forms a fractal curve with a dimension of 7/4.
The observed fractality may provide a characteristic, experimentally-relevant
signature of quantum backflow near the probability-transfer bound.
