  We show unconditional uniqueness of solutions to the Cauchy problem
associated with the Benjamin-Ono equation under the periodic boundary condition
with initial data given in $H^s$ for $s>1/6$. This improves the previous
unconditional uniqueness result in $H^{1/2}$ by Molinet and Pilod (2012). Our
proof is based on a gauge transform and integration by parts in the time
variable.
