## Abstract

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. $\mathbb {P}[{X}_{j}={Y}_{j}] = {\frac{1+\rho }{ 2}}$. 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 $\mathbb {P}[{X} \in {A}, {Y}\in {B}]$.

Original language | American English |
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Article number | 9171899 |

Pages (from-to) | 7878-7886 |

Number of pages | 9 |

Journal | IEEE Transactions on Information Theory |

Volume | 66 |

Issue number | 12 |

DOIs | |

State | Published - Dec 2020 |

### Bibliographical note

Funding Information:Manuscript received September 3, 2019; revised August 3, 2020; accepted August 10, 2020. Date of publication August 20, 2020; date of current version November 20, 2020. The work of Or Ordentlich was supported by ISF under Grant 1791/17. The work of Yury Polyanskiy was supported in part by the National Science Foundation under Grant CCF-17-17842, in part by the Center for Science of Information (CSoI), and in part by the NSF Science and Technology Center under Grant CCF-09-39370. The work of Ofer Shayevitz was supported by the European Research Council under Grant 639573. (Corresponding author: Or Ordentlich.) Or Ordentlich is with the Rachel and Selim Benin School of Computer Science and Engineering, Hebrew University of Jerusalem, Jerusalem 91904, Israel (e-mail: or.ordentlich@mail.huji.ac.il).

Publisher Copyright:

© 1963-2012 IEEE.

## Keywords

- Isoperimetric inequalities
- binary adder multiple access channel (MAC)
- hypercontractivity