a suitable range and analyzed the sum $
\sum_{i=1}^n w_ix_i
$ as $x$ ranged over $\zoon.$ We showed that the subset $X_s\subset\zoon$ that induces any particular sum $s$ modulo $2^{\Theta(n)}$ is relatively large and that membership in $X_s$ has *almost* zero correlation with any given polynomial of low degree. For the purposes of this paper, the correlations with low-degree polynomials need to be *exactly* zero. In this section we show that, with respect to a suitable distribution $\mu_s$ on each $X_s,$ membership in $X_s$ will indeed have zero correlation with any low-degree polynomial.

A starting point in our discussion is a general statement on zeroing out the correlations of given Boolean functions $\chi_1,\chi_2,\dots,\chi_k$ with another Boolean function $f.$ Recall that for functions $f,g\colon X\to\Re$ on a finite set $X,$ we use the notation $$\begin{aligned}
\langle f, g\rangle = \frac1{\abs X} \sum_{x\in X} f(x) g(x).\end{aligned}$$

\[thm:distribution-by-inversion\] Let $f,\chi_1,\dots,\chi_k\colon X\to\moo$ be given functions on a finite set $X.$ Suppose that $$\begin{aligned}
&\sum_{\substack{i=1}}^{k} \abs{\langle f,\chi_i\rangle}
<\frac12,
\label{eqn:f-chi-bounded}\\
&\sum_{\substack{j=1\\j\ne i}}^{k} \abs{\langle \chi_i,\chi_j\rangle} \leq
\frac12, &&i=1,2,\dots,k.
\label{eqn:diag-dominance}\\
\intertext{Then there exists a probability distribution $\mu$ on $X$ such
that}
&\Exp_\mu\,[f(x)\chi_i(x)] = 0, &&i=1,2,\dots,k.
\nonumber\end{aligned}$$

A comment is in order on the hypothesis of Theorem \[thm:distribution-by-inversion\]. The 