Which Wave Numbers Determine the Thermodynamic Stability of Soft Matter Quasicrystals?

For soft matter to form quasicrystals an important ingredient is to have two characteristic length scales in the interparticle interactions. To be more precise, for stable quasicrystals, periodic modulations of the local density distribution with two particular wave numbers should be favored, and the ratio of these wave numbers should be close to certain special values. So, for simple models, the answer to the title question is that only these two ingredients are needed. However, for more realistic models, where in principle all wave numbers can be involved, other wave numbers are also important, specifically those of the second and higher reciprocal lattice vectors. We identify features in the particle pair interaction potentials that can suppress or encourage density modes with wave numbers associated with one of the regular crystalline orderings that compete with quasicrystals, enabling either the enhancement or suppression of quasicrystals in a generic class of systems.