Level-1 inequality
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Taught to me by Zander Kelley; entropic proof originally from Impagliazzo, Moore and Russell.1
The level-1 inequality says that small sets of the hypercube have very little of their weight on degree 1. Let $f : \pmo^n \to \zo$ and $\alpha \ce \E[f]$, then
\[
\W^{1}[f] \le O\p{\alpha^2 \log(1/\alpha)},
\]
which is much smaller than its total weight $\E[f^2] = \alpha$. This makes sense, since we should think of small sets as having high degree (see Hypercontractivity).
Rephrasing
This statement can be proved directly by usual boolean-analytic tools, but it becomes much more beautiful when we normalize $f$ to a relative density $\phi \ce \frac{f}{\alpha}$:
\[
\W^1[\phi] \leq O(\log(1/\alpha)).
\]
In fact, we can rewrite both sides in terms of the corresponding uniform random variable $\BX \sim f^{-1}(1)$:
- The Fourier coefficients of a density function correspond to expectations of the corresponding variable, in this case $\Fou{\phi}(i) \ce \E_{\Bx \sim \pmo^n}[\phi(\Bx)\Bx_i] = \E[\BX_i]$.
- Since $\BX$ is uniform over a set that is a factor $\alpha$ smaller than $\pmo^n$, its entropy deficit is precisely $\D\b{\BX} = \log(1/\alpha)$.
So the level-1 inequality becomes
\[
\sum_{i=1}^n \E[\BX_i]^2 \leq O(\D\b{\BX}).
\]
Note that this rephrasing already makes it obvious that the LHS is $\leq n$, even though the corresponding bound of $\le \alpha^2n$ was not as obvious in the original form.
Proof
Intuitively, the LHS quantifies “how much bias” there is in each coordinate of $\BX$. All we have to show is that if there is a lot of bias in the individual coordinates, then the overall entropy deficit of $\BX$ is large.
And indeed, for any binary variable, entropy deficit is proportional to the squared bias, so we can sum it up and use the superadditivity of entropy deficit:
\[
\al{
\sum_{i=1}^n \E[\BX_i]^2
&\leq \sum_{i=1}^n O(\D\b{\BX_i})\\
&\leq O(\D\b{\BX}).\tag{superadditivity}
}
\]
#to-write If presenting as an application of Entropy deficit, then consider just proving the $\sum_{i=1}^n \E[\BX_i]^2 \leq O(\D\b{\BX})$ form and only vaguely mention the boolean analysis form as a corollary?
-
Russell Impagliazzo, Cristopher Moore, Alexander Russell, “An entropic proof of Chang’s inequality”. ↩