  We performed single electron tunneling measurements on bilayer ruthenate
Sr$_3$Ru$_2$O$_7$. We observe an unusual oscillation in tunneling
magnetoresistance near the metamagnetic quantum phase transition at
temperatures below 7 K. The characteristic features of this oscillation suggest
that it is unrelated to traditional quantum oscillations caused by orbit
quantization. In addition, tunneling spectra are found to change sharply in the
low bias voltage range of $V$ $<$ 2mV near the transition field. These
observations reveal that the Fermi surface of Sr$_3$Ru$_2$O$_7$ changes in a
surprising way as the system undergoes strong critical fluctuations.
