The pivotal set of a Boolean function

TL;DR

This paper introduces the pivotal set of a Boolean function, proves a key inequality about its expected size, and explores related inequalities and formulas, including the Margulis--Russo formula and deviations inequalities.

Contribution

It provides new proofs of an inequality on the pivotal set's expected size and extends classical inequalities to this context, enriching the theoretical understanding.

Findings

Proved a fundamental inequality on the expected size of the pivotal set.

Derived classical and new deviations inequalities for the pivotal set.

Provided multiple proofs of the Margulis--Russo formula.

Abstract

We define the pivotal set of a Boolean function and we prove a fundamental inequality on its expected size, when the inputs are independent random coins of parameter~$p$. We give two complete proofs of this inequality. Along the way, we obtain the classical Margulis--Russo formula. We give a short proof of the classical Hoeffding inequality for i.i.d. Bernoulli random variables, and we use it to derive more complex deviations inequalities associated to the pivotal set. We follow finally Talagrand's footsteps and we discuss a beautiful inequality that he proved in the uniform case.