Theory nt.number – Is the upper Banach density always zero compared to a finite subset sequence?

The following question came to me when I read Hindman and Strauss's paper "Density in arbitrary semigroups".

Question: Given an infinite subset $ A $ of $ mathbb {N} $ such as $ A ^ c $ is also infinite, is there a sequence $ mathcal {F} = {F_n } _ {n in mathbb {N}} $ of finite subset $ mathbb {N} $ so the higher density Banach $ A $, that is to say., $ d _ { mathcal {F}} ^ * (A) = 0 ,? $,

or $ d _ { mathcal {F}} ^ * (A): = sup {alpha: forall n in mathbb {N}, exists , n_0 geq n text {and} x in mathbb {N} text {such as} | A cap (F_ {n_0} + x) | geq alpha | F_ {n_0} | } $.

Even if the answer is negative in general, is it true of certain infinite subsets of $ mathbb {N} ,? $

Thank you in advance for your help.