we use a lower period resolution in the L-S algorithm than what we used to select the variables in Paper II (we sample at $0.1$ times the Nyquist frequency rather than $0.005$ times the Nyquist frequency).

In figure \[fig:RotationRecovery\] we show the recovery fraction as a function of period and of amplitude for the sinusoid injections and the spot model injections while figure \[fig:RotationRecovery\_mag\] shows the recovery fraction as a function of magnitude. When we don’t apply the multi-harmonic AoV selection, the recovery fraction is more or less insensitive to period between $0.1$ and $20$ days for semi-amplitudes larger than $0.01$ mag. The recovery fraction for stars brighter than $r \sim 20$ is close to $100\%$ for the sine-curve model and is between $80-100\%$ for the spot model. The recovery fraction drops at a few periods ($\sim 1~{\rm day}$ and $2-4~{\rm days}$) due to the period rejection that we perform in selecting the variables. For the sinusoid models applying the multi-harmonic AoV selection reduces the recovery fraction dramatically to $10-20\%$. This is due to the degeneracy between harmonic number and period (i.e. doubling the period and setting the amplitude of the first harmonic to one and all other harmonics to 