  Magneto-oscillations in kinetic and magnetic response functions of a few
underdoped cuprates are perhaps one of the most striking observations since
many probes of underdoped cuprates clearly point to a non Fermi-liquid normal
state. Their observation in the vortex state well below the upper critical
field raises a doubt concerning their normal state origin. Here I propose an
explanation of the magneto-oscillations as emerging from the quantum
interference of the vortex lattice and checkerboard modulations of the electron
density of states revealed by STM with atomic resolution in some cuprate
superconductors. The checkerboard effectively pins the vortex lattice, when the
period of the latter is commensurate with the period of the checkerboard. This
condition yields 1/\sqrt{B} periodicity of the response functions versus
magnetic field B, rather than 1/B periodicity of conventional normal state
oscillations. Our solution of the Gross-Pitaevskii-type equation for composed
charged bosons accounting for the d-wave symmetry of the order-parameter and
its checkerboard modulations describes well changes in resonant frequency of
the tunnel-diode oscillator circuit with YBa2Cu4O8 and the oscillatory part of
the Hall resistance and magnetic susceptibility in the mixed state of
YBa2Cu3O6.5.
