cardinality of subset and entropy [closed]

Question

(edited) I am considering the problem about the cardinality of a proper subset, $A\subset\{0,1\}^d$ where $d$ is an integer. Of course, $|A|<2^d$. I am wondering if there is a tighter bound for it. As suggested by Ashok, I suspect the subset cardinality has some thing to do with the entropy, $H(p^*)$ , where $P^*=\sum _{x\in A}P(x)$, and $P(x)$ is the (empirical distribution) type of x. Maybe be the form of $|A|<2^{d*H(P^*)}$, which is similar to the form suggested by Ashok,

any idea will be appreciated. Thanks

Answer 1

I can only think about the cardinality of the typical set $A_\epsilon^{(d)}$ with respect to your distribution on the random variable $X$. If that is what you mean, then it is well known that: \begin{equation} 2^{n(H(X)+\epsilon)}\geq |A_\epsilon^{(d)}|\geq (1-\epsilon)2^{n(H(X)-\epsilon)} \end{equation}