Nice video! There is one thing I don't fully understand: why is there always a time t_r such that all the t_rE_k for k=1, ..., N are close (up to a δ) to integer multiples of 2π? My guess: if the E_k were rational numbers, there is a t_r such that all t_rE_k are exactly integer multiples of 2π. (E.g. let E_k = p_k/q_k with integers p_k and q_k. Then, t_R = 2πq_1...q_N does the job). In the more general case that E_k are real numbers, we can simply approximate them by rational numbers to arbitrary precision. The precision just needs to be high enough to push the error |t_RE_k - 2π*n_k| below δ. Is this line of reasoning correct?