Convergence properties of Fourier mode representations of quasipatterns

Spatial Fourier transforms of quasipatterns observed in Faraday wave experiments suggest that the patterns are well represented by the sum of 8, 10 or 12 Fourier modes with wavevectors equally spaced around a circle. We show that nonlinear interactions of n such Fourier modes generate new modes with wavevectors that approach the original circle no faster than a constant times n¯². These close approaches lead to small divisors in the standard perturbation theory used to compute properties of these patterns, and we show that the convergence of the standard method is questionable in spite of the bound on the small divisors.