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[This post is authored by Gil Kalai, who has kindly “guest blogged” this week’s “open problem of the week”. – T.]

The entropy-influence conjecture seeks to relate two somewhat different measures as to how a boolean function has concentrated Fourier coefficients, namely the total influence and the entropy.

We begin by defining the total influence. Let \{-1,+1\}^n be the discrete cube, i.e. the set of \pm 1 vectors (x_1,\ldots,x_n) of length n. A boolean function is any function f: \{-1,+1\}^n \to \{-1,+1\} from the discrete cube to {-1,+1}. One can think of such functions as “voting methods”, which take the preferences of n voters (+1 for yes, -1 for no) as input and return a yes/no verdict as output. For instance, if n is odd, the “majority vote” function \hbox{sgn}(x_1+\ldots+x_n) returns +1 if there are more +1 variables than -1, or -1 otherwise, whereas if 1 \leq k \leq n, the “k^{th} dictator” function returns the value x_k of the k^{th} variable.

We give the cube \{-1,+1\}^n the uniform probability measure \mu (thus we assume that the n voters vote randomly and independently). Given any boolean function f and any variable 1 \leq k \leq n, define the influence I_k(f) of the k^{th} variable to be the quantity

I_k(f) := \mu \{ x \in \{-1,+1\}^n: f(\sigma_k(x)) \neq f(x) \}

where \sigma_k(x) is the element of the cube formed by flipping the sign of the k^{th} variable. Informally, I_k(f) measures the probability that the k^{th} voter could actually determine the outcome of an election; it is sometimes referred to as the Banzhaf power index. The total influence I(f) of f (also known as the average sensitivity and the edge-boundary density) is then defined as

I(f) := \sum_{k=1}^n I_k(f).

Thus for instance a dictator function has total influence 1, whereas majority vote has total influence comparable to \sqrt{n}. The influence can range between 0 (for constant functions +1, -1) and n (for the parity function x_1 \ldots x_k or its negation). If f has mean zero (i.e. it is equal to +1 half of the time), then the edge-isoperimetric inequality asserts that I(f) \geq 1 (with equality if and only if there is a dictatorship), whilst the Kahn-Kalai-Linial (KKL) theorem asserts that I_k(f) \gg \frac{\log n}{n} for some k. There is a result of Friedgut that if I(f) is bounded by A (say) and \varepsilon > 0, then f is within a distance \varepsilon (in L^1 norm) of another boolean function g which only depends on O_{A,\varepsilon}(1) of the variables (such functions are known as juntas).

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