On hypercube statistics

Let $d \geq 1$ and $s \leq 2^d$ be nonnegative integers. For a subset $A$ of vertices of the hypercube $Q_n$ and $n\geq d$, let $\lambda(n,d,s,A)$ denote the fraction of subcubes $Q_d$ of $Q_n$ that contain exactly $s$ vertices of $A$. Let $\lambda(n,d,s)$ denote the maximum possible value of $\lambda(n,d,s,A)$ as $A$ ranges over all subsets of vertices of $Q_n$, and let $\lambda(d,s)$ denote the limit of this quantity as $n$ tends to infinity. We prove several lower and upper bounds on $\lambda(d,s)$, showing that for all admissible values of $d$ and $s$ it is larger than $0.28$. We also show that the values of $s=s(d)$ such that $\lambda(d,s)=1$ are exactly $\{0,2^{d-1},2^d\}$. In addition we prove that if $0<s< d/8$, then $\lambda(d, s) \leq 1 - \Omega(1/s)$, and that if $s$ is divisible by a power of $2$ which is $\Omega(s)$ then $\lambda(d,s) \geq 1-O(1/s)$. We suspect that $\lambda(d,1)=(1+o(1))/e$ where the $o(1)$-term tends to $0$ as $d$ tends to infinity, but this remains open, as does the problem of obtaining tight bounds for essentially all other quantities $\lambda(d,s)$.