A Note on the Probability of Rectangles for Correlated Binary Strings

Consider two sequences of n independent and identically distributed fair coin tosses, X = (X
₁
, . . . , X
_n
) and Y = (Y
₁
, . . . , Y
_n
), which are ρ-correlated for each j, i.e. P[X
_j
= Y
_j
] = 1+ρ/2 .We study the question of how large (small) the probability P[X ∈ A, Y ∈ B] can be among all sets A, B ⊂ {0, 1}
ⁿ
of a given cardinality. For sets |A|, |B| = Θ(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
^Θ
(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 P[X ∈ A, Y ∈ B] in the regime of p → 1. We also prove a similar tight lower bound, i.e. show that for p → 0 the pair of opposite Hamming balls approximately minimizes the probability P[X ∈ A, Y ∈ B].