# 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 $$mathbb {N} ,?$$