Bounds for the number of Boolean functions admitting affine approximations of a given accuracy

Abstract

We obtain two-sided bounds and asymptotic formulas for the number of Boolean functions of n variables which are approximated by affine or linear Boolean functions with a given accuracy.

Exact value for subgaussian norm of centered indicator random variable

“…Thus, the estimate (3.3) is non-improvable; cf. [1], [11], [14], [15]. Another generalizations of the equality (2.2), for example, on the Hoeffding's inequality and on the theory of martingales see in the article of M.Raginsky and I.Sason [12].…”

Limit Distribution of the Hamming Distance from the Random Boolean Function to the Set of Affine Functions

“…Theorem 1, as the result of Ryasanov [5], concerns the "central" parts of distributions of ρ(f, An) and ρ(f, Ln). It was proved in [7] that in the domains of "large deviations" (when x → −∞ as n → ∞) the right-hand sides of (1) and (2) are incorrect asymptotics of probabilities in their left-hand sides.…”