  A function $f\colon\{0,1\}^n\to \{0,1\}$ is called an approximate
AND-homomorphism if choosing ${\bf x},{\bf y}\in\{0,1\}^n$ randomly, we have
that $f({\bf x}\land {\bf y}) = f({\bf x})\land f({\bf y})$ with probability at
least $1-\epsilon$, where $x\land y = (x_1\land y_1,\ldots,x_n\land y_n)$. We
prove that if $f\colon \{0,1\}^n \to \{0,1\}$ is an approximate
AND-homomorphism, then $f$ is $\delta$-close to either a constant function or
an AND function, where $\delta(\epsilon) \to 0$ as $\epsilon\to0$. This
improves on a result of Nehama, who proved a similar statement in which
$\delta$ depends on $n$.
  Our theorem implies a strong result on judgement aggregation in computational
social choice. In the language of social choice, our result shows that if $f$
is $\epsilon$-close to satisfying judgement aggregation, then it is
$\delta(\epsilon)$-close to an oligarchy (the name for the AND function in
social choice theory). This improves on Nehama's result, in which $\delta$
decays polynomially with $n$.
  Our result follows from a more general one, in which we characterize
approximate solutions to the eigenvalue equation $\mathrm T f = \lambda g$,
where $\mathrm T$ is the downwards noise operator $\mathrm T f(x) =
\mathbb{E}_{{\bf y}}[f(x \land {\bf y})]$, $f$ is $[0,1]$-valued, and $g$ is
$\{0,1\}$-valued. We identify all exact solutions to this equation, and show
that any approximate solution in which $\mathrm T f$ and $\lambda g$ are close
is close to an exact solution.
