# Approximation of biased Boolean functions of small total influence by DNF's

## TL;DR

This paper proves that biased Boolean functions with small total influence can be closely approximated by small DNF formulas, extending the Junta theorem to biased functions and providing nearly optimal bounds.

## Contribution

It establishes a structure theorem showing biased functions with near-minimal influence can be approximated by small DNFs, answering a question by Kahn and Kalai.

## Key findings

- Functions with influence close to the minimum can be approximated by small DNFs.
- The size of the approximating DNF depends double-exponentially on the influence parameter.
- The bounds on DNF size are nearly optimal up to constants.

## Abstract

The influence of the $k$'th coordinate on a Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$ is the probability that flipping $x_k$ changes the value $f(x)$. The total influence $I(f)$ is the sum of influences of the coordinates. The well-known `Junta Theorem' of Friedgut (1998) asserts that if $I(f) \leq M$, then $f$ can be $\epsilon$-approximated by a function that depends on $O(2^{M/\epsilon})$ coordinates. Friedgut's theorem has a wide variety of applications in mathematics and theoretical computer science. For a biased function with $E[f]=\mu$, the edge isoperimetric inequality on the cube implies that $I(f) \geq 2\mu \log(1/\mu)$. Kahn and Kalai (2006) asked, in the spirit of the Junta theorem, whether any $f$ such that $I(f)$ is within a constant factor of the minimum, can be $\epsilon \mu$-approximated by a DNF of a `small' size (i.e., a union of a small number of sub-cubes). We answer the question by proving the following structure theorem: If $I(f) \leq 2\mu(\log(1/\mu)+M)$, then $f$ can be $\epsilon \mu$-approximated by a DNF of size $2^{2^{O(M/\epsilon)}}$. The dependence on $M$ is sharp up to the constant factor in the double exponent.