Discrete correlations of order 2 of generalised Rudin–Shapiro sequences: a combinatorial approach

The paper defines generalised Rudin–Shapiro sequences via a difference matrix and proves two main results: (i) the discrepancy bound 1/N |#{0 ≤ n < N: u_{n+r} − u_n = g} − 1/|G|| ≤ rk(1+log_k N)/N (Theorem 2.10), and (ii) the limiting joint distribution lim_{N→∞} (1/N) #{0 ≤ n < N: (u_n,u_{n+r})=(i,j)} = 1/|G|^2 (Theorem 2.12). Both statements and their proofs are clearly presented and rely on a combinatorial “fibre” argument plus uniform letter frequencies for primitive morphic sequences . By contrast, the candidate solution’s Fourier/packet-cancellation approach makes a crucial, unjustified cancellation: it invokes “row orthogonality” of the character matrix R_χ (which follows from the difference-matrix condition) but then sums over the row index (varying x_L) with a fixed column h. The difference-matrix property yields orthogonality when summing over columns h for fixed rows i≠j, not when summing over rows with a fixed column. Hence the asserted top-level packet cancellation, and the O(rk) remainder bound that follows, are not justified. The paper’s results (Propositions 3.3–3.4 for (i), Proposition 3.7 for (ii)) are correct and complete using the fibre method .