  Consider two sequences of $n$ independent and identically distributed fair
coin tosses, $X=(X_1,\ldots,X_n)$ and $Y=(Y_1,\ldots,Y_n)$, which are
$\rho$-correlated for each $j$, i.e. $\mathbb{P}[X_j=Y_j] = {1+\rho\over 2}$.
We study the question of how large (small) the probability $\mathbb{P}[X \in A,
Y\in B]$ can be among all sets $A,B\subset\{0,1\}^n$ of a given cardinality.
For sets $|A|,|B| = \Theta(2^n)$ it is well known that the largest (smallest)
probability is approximately attained by concentric (anti-concentric) Hamming
balls, and this can be proved via the hypercontractive inequality (reverse
hypercontractivity). Here we consider the case of $|A|,|B| = 2^{\Theta(n)}$. By
applying a recent extension of the hypercontractive inequality of
Polyanskiy-Samorodnitsky (J. Functional Analysis, 2019), we show that Hamming
balls of the same size approximately maximize $\mathbb{P}[X \in A, Y\in B]$ in
the regime of $\rho \to 1$. We also prove a similar tight lower bound, i.e.
show that for $\rho\to 0$ the pair of opposite Hamming balls approximately
minimizes the probability $\mathbb{P}[X \in A, Y\in B]$.
