Polynomial mean complexity and logarithmic Sarnak conjecture

In this paper, we reduce the logarithmic Sarnak conjecture to the {0, 1}-symbolic systems with polynomial mean complexity. By showing that the logarithmic Sarnak conjecture holds for any topologically dynamical system with sublinear complexity, we provide a variant of the 1-Fourier uniformity conjecture, where the frequencies are restricted to any subset of [0, 1] with packing dimension less than one.