  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.
