Junta

A $J$-junta is a function whose value depends only on coordinates indexed by $J$. A $k$-junta depends on at most $k$ coordinates.

If $f:\{-1,1{\}}^n\to \{-1,1\}$ satisfies $\Vert f^{\le k}{\Vert }_2^2\ge 1-\epsilon$, then it has a real-valued $J$-junta approximation $g$ with $\Vert f-g{\Vert }_2^2\le 2\epsilon$ and

For every $\epsilon >0$, a Boolean function $f$ has an $\exp (O(\mathbf{I}(f)/\epsilon ))$-junta approximation with squared $L^2$ error at most $2\epsilon$.

Every Boolean function of degree at most $k$ is a $k{2}^{k-1}$-junta.