Explore a lecture on the convergence of planewave approximations for quantum incommensurate systems. Delve into the numerical approximations of spectrum distribution for Schrödinger operators in incommensurate systems, focusing on the density of states. Examine the thermodynamic limit justification, planewave approximation methods with novel energy cutoffs, and convergence analysis with error estimates. Discover an efficient algorithm for evaluating density of states through reciprocal space sampling. Follow the progression from periodic structures to incommensurate systems, exploring Schrödinger-type eigenvalue problems, band structures, and matrix structures. Investigate supercell approximations, planewave discretizations, and higher-dimensional formulations. Conclude with practical examples demonstrating the convergence of planewave approximations and a comprehensive summary of the topic.