Following , the number of samples per wavelength I will call . A practical choice of will depend on what type of wavefunction and billiard we have, as well as the required accuracy of .
I found is sufficient for most work involving the stadium billiard and evanescent waves components of or less. In this shape, the convergence with does not follow any obvious power-law, but for I found the fraction error on is for wavefunctions composed of random (real) plane waves. If highly evanescent waves are present (up to ), needs to be slightly higher (about 10). Certainly the of 30 quoted by  is excessive, considering the fact that their PWDM basis set only includes real plane waves. In the scaling method, I find that errors due to a smaller cause spurious states (Section 6.3.3) to enter earlier than is necessary. However, the accuracy of the valid states found does not seem very dependent on .
Further study on the accurate evaluation of these integrals is necessary; in particular a way to handle corners and `kinks' which has high-order (preferably exponential) convergence is sought.