Core Concepts

Every Boolean function on n variables admits a d-set of variables with influence at least 1/10 * W≥d(f) * (log n/n)^d, which generalizes the celebrated Kahn-Kalai-Linial theorem.

Abstract

The content discusses the notion of the influence of a set of variables on a Boolean function, which was recently introduced by Tal. The main result is a generalization of the Kahn-Kalai-Linial (KKL) theorem, which states that every Boolean function on n variables admits a very influential single bit.
The key points are:
The author introduces the notion of the influence of a set of variables i on a Boolean function f, denoted as Inf_i(f), which measures if all the bits in i are influential on f.
The author proves that for every Boolean function f on n variables and any fixed d, there exists a d-set of variables i such that Inf_i(f) ≥ 1/10 * W≥d(f) * (log n/n)^d, where W≥d(f) is the Fourier mass of f at levels greater than or equal to d. This generalizes the KKL theorem, which corresponds to the case d = 1.
The author also presents a family of "d-hypertribe" functions that demonstrate the essential sharpness of this result for every fixed d.
Additionally, the author proves a d-degree generalization of a related theorem by Oleszkiewicz, showing that if all (d+1)-degree influences of a Boolean function are small, then the function is close to some d-degree Boolean function.
The proofs rely on tools from harmonic analysis on the Boolean cube, such as the Fourier-Walsh expansion, partial derivatives, and hypercontractivity.

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Deeper Inquiries

The techniques and insights from the work on the influence of sets of variables in the Boolean cube can be applied to study similar concepts in other discrete domains. For instance, in graph theory, one could investigate the influence of sets of vertices or edges on certain graph properties or functions. By adapting the Fourier analysis techniques used in the Boolean cube context, one could analyze the structural properties of graphs based on the influence of specific subsets of vertices or edges. This could lead to a deeper understanding of how different parts of a graph contribute to its overall behavior or properties.

The d-degree generalization of the Oleszkiewicz theorem has significant implications for the structure of Boolean functions with small high-degree Fourier coefficients. This generalization allows for a more nuanced analysis of the influences of sets of variables on Boolean functions. By considering influences at different degrees, one can gain insights into how different levels of interactions between variables affect the behavior of the function. This can lead to a better understanding of the complexity and structure of Boolean functions with specific Fourier coefficients at higher degrees, providing a more detailed characterization of their properties.

The d-hypertribe construction can be further generalized to provide sharp examples for other notions of influence or higher-order interactions in Boolean function analysis. By adapting the construction to focus on different types of influences or interactions, such as pairwise interactions or interactions among larger sets of variables, one can create families of functions that demonstrate the sharpness of various theorems or results in Boolean function analysis. This generalization can help explore the relationships between different types of influences and the overall behavior of Boolean functions, offering insights into the complexity and structure of these functions in various contexts.

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