A critical Poisson law in a dyadic block

Let f be an extended Rademacher random multiplicative function, i.e. a completely multiplicative function whose values at the primes are independent Rademacher variables. We count, in the dyadic block [N,2N), the starting points of maximal constant stretches of length at least L. Uniformly for |L−log₂N| ≤ C, we prove that this count is Poisson in total variation with parameter N·2^(−L), at rate O_C((log log N)^(−2)); the dominant term comes from the probabilistic conditioning, while the arithmetic core leaves an exp(−c·√(log N)/log log N) margin. The proof combines an exact affine formulation, an absolute homogeneous two-block estimate, conditioning on the small primes, and the Chen–Stein method. We also prove a uniform deterministic-mask version, vague convergence of the spatial start process to a Poisson point process, and convergence of the excess-length marked process to a Poisson point process with geometric marks. The gluing of [1,M] is proved for interior starts, and the longest-run law in the finite prefix (f(1),…,f(M)) follows by coupling.