If \(X\) is \(B\)-reasonable and not equivalent to \(0\) (its second moment is positive), then for every \(t \gt 0\),

Let \(X\) be a discrete random variable on a finite space whose law is a PMF \(\pi \). If \(\mu \gt 0\) is a lower bound on every atom probability \(\pi (\omega )\), then \(X\) is \((1/\mu )\)-reasonable.

Let \(Z \ge 0\) be a nonnegative random variable with finite first and second moments and positive mean. For every \(\theta \) with \(0 \le \theta \le 1\),

If \(X\) is \(B\)-reasonable and not equivalent to \(0\), then for every \(t \in [0,1]\),

For every \(x\), the restriction of \(f\) fixing the last coordinate to false equals the sum of the average and the half-difference: \(\mathrm{restrictLast}\, f\, \mathrm{false}\, x = \mathrm{avgLast}\, f\, x + \mathrm{diffLast}\, f\, x\).

For every \(x\), the restriction of \(f\) fixing the last coordinate to true equals the difference of the average and the half-difference: \(\mathrm{restrictLast}\, f\, \mathrm{true}\, x = \mathrm{avgLast}\, f\, x - \mathrm{diffLast}\, f\, x\).

For any \(\varphi \) on the cube of dimension \(n+1\), the total sum decomposes according to the value of the last coordinate:

The uniform point weight satisfies \(\mathrm{uniformWeight}(n+1) = \mathrm{uniformWeight}(n)/2\).

For each \(S \subseteq \mathrm{Fin}\, n\), the Fourier coefficient of \(\mathrm{avgLast}\, f\) at \(S\) equals the Fourier coefficient of \(f\) at the image of \(S\) under castSucc: \(\widehat{\mathrm{avgLast}\, f}(S) = \widehat{f}(\mathrm{castSucc}(S))\).

For each \(S \subseteq \mathrm{Fin}\, n\), the Fourier coefficient of \(\mathrm{diffLast}\, f\) at \(S\) equals the Fourier coefficient of \(f\) at the image of \(S\) under castSucc together with the last index: \(\widehat{\mathrm{diffLast}\, f}(S) = \widehat{f}\bigl(\mathrm{castSucc}(S) \cup \{ \mathrm{last}\, n\} \bigr)\).

The expectation of \(\varphi \) on the cube of dimension \(n+1\) is the average of the expectations of its two restrictions:

For all real \(a, b\), \((a+b)^4 + (a-b)^4 = 2\, (a^4 + 6\, a^2 b^2 + b^4)\).

For all real \(a, b\), \((a+b)^2 + (a-b)^2 = 2\, (a^2 + b^2)\).

Writing \(g = \mathrm{avgLast}\, f\) and \(h = \mathrm{diffLast}\, f\), the fourth moment of \(f\) decomposes as

Writing \(g = \mathrm{avgLast}\, f\) and \(h = \mathrm{diffLast}\, f\), the second moment of \(f\) decomposes as \(\mathbb {E}[f^2] = \mathbb {E}[g^2] + \mathbb {E}[h^2]\).

For Boolean functions \(g, h\) on the cube, \(\bigl(\mathbb {E}[g^2 h^2]\bigr)^2 \le \mathbb {E}[g^4]\cdot \mathbb {E}[h^4]\).

For every Boolean function \(f\), \(0 \le \mathbb {E}[f^2]\).

For Boolean functions \(g, h\), \(0 \le \mathbb {E}[g^2 h^2]\).

For every Boolean function \(f\), \(0 \le \mathbb {E}[f^4]\).

If \(f\) has degree at most \(k\), then its average over the last coordinate \(\mathrm{avgLast}\, f\) also has degree at most \(k\).

If \(f\) has degree at most \(k\), then its half-difference over the last coordinate \(\mathrm{diffLast}\, f\) has degree at most \(k-1\).

A Boolean function \(f\) of degree at most \(0\) is constant: \(f(x) = f(\mathrm{default})\) for all \(x\).

For a Boolean function \(f\) of degree at most \(0\) (hence constant), \(\mathbb {E}[f^4] = \bigl(\mathbb {E}[f^2]\bigr)^2\).

The key algebraic inequality driving the inductive step of the Bonami lemma: for \(a, b, A, B, C \ge 0\) (with \(a, b, B, C\) nonnegative) satisfying \(A \le 9^{m+1} a^2\), \(B \le 9^m b^2\), and \(C^2 \le A\cdot B\), one has

The main Bonami inequality stated through the expectation operator: if a Boolean function \(f\) has degree at most \(k\), then

If \(P\) is a probability measure on the cube assigning each point the uniform weight, then the \(p\)-th moment of \(f\) under \(P\) coincides with the expectation \(\mathbb {E}[f^p]\).

The uniform measure assigns each point \(x\) of the cube the combinatorial uniform weight: \((\mathrm{uniformMeasure}\, n)(\{ x\} ) = \mathrm{uniformWeight}\, n\).

A Boolean function \(f\) of degree at most \(k\) is \(9^{k}\)-reasonable under the uniform measure on the hypercube.