Example of a Complex Function with Bounded Influence and High Entropy

Core Concepts

There exists a complex function f from {-1, 1}^n to the unit sphere in C with influence bounded by 1 and entropy of |f̂|^2 larger than (1/2) log n.

Abstract

The content presents an example of a complex function f from the Cantor group {-1, 1}^n to the complex plane C, with modulus 1, that has bounded influence but high entropy.
The key highlights are:

The function F is defined as (1 + 1/n)^(-n/2) * Π(1 + iε_j/√n), where ε_j are the projections onto the j-th coordinate.
The influence I(F) of this function is calculated to be n/(n+1), which is bounded by 1.
The entropy H(|F̂|^2) of the Fourier coefficients of F is shown to be greater than (n log n)/(n+1), which is larger than (1/2) log n for large n.
This provides a counterexample to the analogous version of the Friedgut-Kalai entropy-influence conjecture for complex functions of modulus one, as posed by Gady Kozma.
The author notes that this simple example was likely known to experts, but not published before.

Stats

I(F) = n/(n+1)
H(|F̂|^2) > (n log n)/(n+1)

Quotes

None

Key Insights Distilled From

Entropy versus influence for complex functions of modulus one 2

by Gideon Schec... at arxiv.org 04-25-2024

https://arxiv.org/pdf/2009.12753.pdf

Deeper Inquiries

How can the construction of this example be generalized to other classes of complex functions?

The construction of the example provided in the context can be generalized to other classes of complex functions by considering functions with modulus one that exhibit similar properties. One way to generalize this construction is to explore functions that have specific properties related to their Fourier coefficients and entropy. By manipulating the coefficients and structure of the functions, it is possible to create a variety of examples that showcase different relationships between influence and entropy for complex-valued functions. This generalization can lead to a deeper understanding of the interplay between these two concepts in the context of complex functions.

What are the implications of this counterexample for the study of influence and entropy in the context of complex-valued functions?

The counterexample presented in the context has significant implications for the study of influence and entropy in the context of complex-valued functions. It demonstrates that the relationship between influence and entropy for complex functions with modulus one can differ from the corresponding relationship for boolean functions. This highlights the complexity and richness of the behavior of complex functions and suggests that traditional results and conjectures from the boolean setting may not directly apply to the complex setting. By studying such counterexamples, researchers can gain insights into the unique properties and characteristics of complex functions, leading to a more nuanced understanding of their behavior in relation to influence and entropy.

What other properties of complex functions, beyond influence and entropy, could be investigated to gain a deeper understanding of their behavior?

Beyond influence and entropy, several other properties of complex functions could be investigated to gain a deeper understanding of their behavior. One such property is the concept of correlation decay, which measures how quickly correlations between different coordinates of the input space decay as the distance between the coordinates increases. Studying correlation decay in complex functions can provide insights into their structure and interactions. Additionally, properties related to the smoothness, regularity, and analyticity of complex functions can be explored to understand their behavior in different contexts. Investigating the behavior of complex functions under various transformations and operations, such as composition, convolution, and inversion, can also shed light on their intrinsic properties and relationships. By delving into these and other properties, researchers can uncover the rich and diverse nature of complex functions and deepen their understanding of these mathematical objects.