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 Incoming links: Entropy method for certifiable functions

Exactly one lowest-numbered ball is selected in each occupied bin, so

\sum_{j = 1}^{m} α_{j} (x) = n - f (x) \leq n, \sum_{j = 1}^{m} α_{j} (x)^{2} \leq n .

Part c supplies the corresponding one-sided coordinate certificate. The product-space entropy method for certifiable functions states that a function with such a certificate of squared size at most

v

has both centered tails bounded by

e^{- t^{2} / (2 v)}

. Taking

v = n

gives

P (Z - E Z \geq t) \leq e^{- t^{2} / (2 n)}, P (Z - E Z \leq - t) \leq e^{- t^{2} / (2 n)} .

The lower-tail form of the entropy method for certifiable functions states that a unit-bounded-difference,

g

-certifiable nonnegative integer-valued function satisfies

lo g E e^{- λ (Z - E Z)} \leq \frac{λ ^{2}}{2} E [g (Z)] (λ \geq 0) .

It follows by applying entropy tensorization to a minimal certificate: only its at most

g (Z)

coordinates can contribute to the one-sided variance proxy, and changing any one contributes at most one.

The Chernoff bound therefore gives

P (Z - E Z \leq - t) \leq in f_{λ > 0} exp (- λ t + \frac{λ ^{2}}{2} E [g (Z)]) .

Choosing

λ = t / E [g (Z)]

proves

P (Z - E Z \leq - t) \leq exp (- \frac{t ^{2}}{2 E [ g ( Z )]}) .

Tensorization of entropy Created 2026-09-24 Updated 2026-09-24

For a nonnegative function

Z

of independent random variables, entropy is at most the sum of its conditional coordinate entropies. This tensorization is a basic step of the entropy method.