nonsingular by Theorem \[thm:gershgorin\]. Fix the unique solution $\alpha$ to the system (\[eqn:lin-system\]). Then $2\abs{\alpha_i} - \sum_{j=1}^{k} \abs{\alpha_j\langle
\chi_i,\chi_j\rangle} \leq \abs{\langle f,\chi_i\rangle}$ for $i=1,2,\dots,k.$ Summing these $k$ inequalities, we obtain $$\begin{aligned}
2\sum_{i=1}^k \abs{\alpha_i} - \sum_{j=1}^{k}\abs{\alpha_j}
\sum_{i=1}^k\abs{\langle \chi_i,\chi_j\rangle} \leq
\sum_{i=1}^k\abs{\langle f, \chi_i\rangle}, \end{aligned}$$ which in view of (\[eqn:f-chi-bounded\]) and (\[eqn:diag-dominance\]) shows that $\sum_{i=1}^k\abs{\alpha_i}<1.$ Therefore, the function $\mu\colon X\to\Re$ given by $$\begin{aligned}
\mu(x) = \epsilon{\left(
 1 - f(x)\sum_{i=1}^k \alpha_i \chi_i(x)
 \right)}\end{aligned}$$ is a probability distribution on $X$ for a suitable normalizing factor $\epsilon>0.$ At last, $$\begin{aligned}
\Exp_{\mu}\,[f(x)\chi_i(x)] =
\epsilon \abs{X} \left(
\langle f,\chi_i\rangle - \sum_{j=1}^k \alpha_j\langle
\chi_i,\chi_j\rangle \right) = 0,\end{aligned}$$ where the final equality holds by (\[eqn:lin-system\]).

We are now in a position to prove the main result of this section.

\[thm:mu\_s-constructed\] Let $\alpha>0$ be a sufficiently small absolute constant. Choose integers $w_1,w_2,\dots,w_n$ uniformly at random from $\{0,1,\dots,2^{\lfloor\alpha n\rfloor+1}-1\}.$ For $s\in\ZZ,$ define $$\begin{aligned}
\label{eqn:X_s}
X_s = \left\{
 x\in\zoon :  \sum_{i=1}^n w_ix_i \equiv s \pmod{2^{\lfloor\alpha
 n\rfloor+1}}\;
     \right\}. \end{aligned}$$ Then with probability at least $1-\e^{-n/3}$ over the choice of $w_1,w_2,\dots,w_n,$ there is a distribution $\mu_s$ on $X_s$ $($for each $s)$ such that $$\begin{aligned}
\label{eqn:mu_s-equivalence}
\Exp_{\mu_s}\, [p(x)] = \Exp_{\mu_t}\, [p(x)]\end{aligned}$$ for any $s,t\in\ZZ$ and any polynomial $p$ of degree at most $\lfloor \alpha n\rfloor.$

Let 