only a *constant* number of inputs bits. To see this, we view the bijection ${\psi}\colon{\{0,1\}^{n}}\to{\mathcal{B}_{n}}$ as a vector of Boolean functions $\langle {\psi}_1,\ldots, {\psi}_{n+1}\rangle$, where ${\psi}_i(x)$ is the $i^{\text{th}}$ output bit of ${\psi}$ on input $x\in{\{0,1\}^{n}}$. Recall that the *total influence* of a Boolean function ${\psi}_i \colon {\{0,1\}^{n}}\to\{0,1\}$ is the quantity $${\mathbf{Inf}}[{\psi}_i] =  {\mathop{{\bf E}\/}}_{x\sim {\{0,1\}^{n}}}\left[\#\{j\in [n] \colon {\psi}_i(x) \ne {\psi}_i(x+e_j)\}\right].$$ By linearity of expectation, Theorem \[thm:main\] implies that a typical ${\psi}_i$ has bounded total influence. $${\mathop{{\bf E}\/}}_{i \sim [n+1]} [{\mathbf{Inf}}[{\psi}_i]] = {{\sf avgStretch}}({\psi}) \le {{\sf maxStretch}}({\psi}) \le 4. \label{small-total-influence}$$ Next we recall Friedgut’s Junta Theorem, which states that a Boolean function with bounded total influence is well-approximated by another Boolean function that only depends on a *constant* number of input bits. More precisely,

Let $f \colon {\{0,1\}^{n}}\to\{0,1\}$ be a Boolean function. For every ${\varepsilon}> 0$ there exists a Boolean function $g \colon \{0,1\}^n \to\{0,1\}$ such that $g$ is a $2^{O({\mathbf{Inf}}[f]/{\varepsilon})}$-junta[^4] and ${\mathop{\bf Pr\/}}[f(x) \ne g(x)] \le {\varepsilon}$.

Combining Equation  with Friedgut’s Junta Theorem, we see that for any constants $\delta,{\varepsilon}> 0$, all but a $\delta$-fraction of the ${\psi}_i$’s are ${\varepsilon}$-approximated by $O(1)$-juntas. In fact, we give a direct proof that the function ${\psi}$ in Theorem \[thm:main\] satisfies the following 