The largest support to escape chaos in random multiplicative functions

Let f ( n ) be a Steinhaus random multiplicative function and let A ⊂ [ 1 , N ] be a finite set of integers. We show that convergence of | A | − 1 / 2 ∑ n ∈ A f ( n ) to a standard complex normal distribution forces | A | = o ( N ) . We prove that this zero-density condition is sharp: For most sets A of density ρ , and hence for some such sets, a complex Gaussian limit holds whenever ( 1 − ρ ) − 1 = o ( ( log log N ) 1 / 2 ) . However, for positive density, the correct normalization is ( 1 − ρ ) | A | rather than | A | . Thus, the additional factor 1 − ρ is nontrivial precisely when the density does not tend to zero.