Specifically, we define an appropriate distribution on $n$-tuples $(\alpha_1,\dots,\alpha_n)$ and study the behavior of the sum $\sum\alpha_ix_i$ as the vector $x$ ranges over $\zoon.$ We prove that for a typical $n$-tuple $(\alpha_1,\dots,\alpha_n$) and any collection of sums $S\subset\ZZ$ of interest, the subset $X_S\subset\zoon$ that induces the sums in $S$ is highly random in that membership in $X_S$ is uncorrelated with any polynomial of degree up to $\Theta(n).$ With some additional work, this allows the sought passage to a univariate question. In particular, we are able to prove the existence of a halfspace $f\colon\zoon\to\moo$ such that any multivariate rational approximant for $f$ gives a univariate rational approximant for the sign function on $\{\pm1,\pm2,\pm3,\dots,\pm2^{\Theta(n)}\}$ with the same degree and error. The univariate question being well-understood, we infer that $f$ requires a rational function of degree $\Omega(n)$ for pointwise approximation within $1/3$ and hence $\degthr(f\wedge
f)\geq\Omega(n)$ by the characterization from [@sherstov09hshs].

Preliminaries {#sec:prelim}
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Notation. {#notation. .unnumbered}
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We will view Boolean functions as mappings $X\to\zoo$ or $X\to\moo$ for some finite set $X,$ where the output value $1$ corresponds to “true” in the former case and “false” in the latter. We adopt the following standard definition of the sign function: $$\begin{aligned}
  \sign x 