Green's Open Problem #55: Correlation Conjecture

Hey guys! Today, we're diving deep into a fascinating problem posed by Ben Green, specifically Open Problem #55. This conjecture sits at the intersection of several exciting areas, including Fourier analysis, additive combinatorics, and theoretical computer science. Let's break it down and explore its significance.

Unpacking Green's Open Problem #55

At its heart, Green's Open Problem #55 deals with the correlation of functions over finite fields. To fully grasp this, we need to dissect the conjecture piece by piece.

• The Setup: We start with an odd prime number, denoted by p. Consider a function f that maps pairs of elements from the vector space _p^n (the n-dimensional vector space over the finite field with p elements) to the complex numbers \mathbb{C}. A crucial constraint is that this function f is bounded pointwise by 1; in other words, the absolute value of f(x, y) is always less than or equal to 1.
• The Condition: Next, we have a condition involving the Gowers uniformity norm. Specifically, it involves the fourth power of the Gowers uniformity norm of a certain difference of the function f. The expression _h || \Delta{(h,h)} f ||_{\square}^4 \geq \delta essentially states that a certain average involving the function f and its shifts is bounded below by some positive quantity \delta. Here, \Delta{(h,h)} f(x, y) = f(x + h, y + h) - f(x, y). Intuitively, this condition is saying something about how much f fails to be uniform.
• The Question: The central question is whether, under the given condition, the function f correlates with a function of a specific form: a(x)b(y)c(x+y)(-1)^{q(x,y)}. Let's break down what this form means:
  • a(x), b(y), and c(x+y) are functions that depend only on x, y, and x+y, respectively.
  • q(x, y) is a quadratic form in x and y. A quadratic form is a homogeneous polynomial of degree 2. The term (-1)^{q(x,y)} introduces a quadratic phase.

In essence, the conjecture asks: If f exhibits a certain lack of uniformity (as measured by the Gowers norm condition), must it necessarily have a noticeable correlation with a function that can be expressed as a product of functions depending on x, y, and x+y, modulated by a quadratic phase?

Why is this Conjecture Interesting?

This conjecture is compelling for several reasons:

1. Connections to Additive Combinatorics: The problem is deeply rooted in additive combinatorics, which studies the structure of sets and functions under addition. The Gowers uniformity norm is a central tool in this area, used to detect arithmetic patterns in sets and functions. Understanding the behavior of functions with bounded Gowers norms is crucial for advancing our knowledge of additive structures.
2. Links to Theoretical Computer Science: Correlation is a fundamental concept in theoretical computer science, especially in areas like circuit complexity and pseudorandomness. Understanding which functions correlate with certain types of functions is essential for proving lower bounds and designing efficient algorithms. The quadratic phase factor (-1)^{q(x,y)} is of particular interest because quadratic forms play a significant role in various computational problems.
3. Generalization of Existing Results: This conjecture can be seen as a generalization of known results in Fourier analysis and additive combinatorics. For instance, it builds upon the work of Green and others on the structure of sets with small sumsets. A proof of this conjecture would likely lead to new insights and techniques applicable to a broader range of problems.

The Significance of Each Component

Let's delve deeper into why each component of the conjecture is significant.

• The Function f: The function f being bounded pointwise by 1 is a normalization condition. It ensures that we are dealing with functions whose values do not grow too large, which simplifies the analysis. Without this condition, the problem would become significantly more complicated.
• The Gowers Norm Condition: The condition involving the Gowers uniformity norm is the heart of the conjecture. The Gowers norm measures the uniformity of a function. A function with a large Gowers norm is highly non-uniform and exhibits strong arithmetic patterns. The fact that _h || \Delta{(h,h)} f ||_{\square}^4 \geq \delta tells us that f has some inherent structure that prevents it from being completely random.
• The Correlation Question: The question of whether f correlates with a function of the form a(x)b(y)c(x+y)(-1)^{q(x,y)} is the crux of the problem. Correlation, in this context, means that the inner product of f with the given function is non-negligible. In other words, there is a statistical dependence between f and the specified form. If f correlates with this form, it suggests that f has a specific type of algebraic structure.
• The Target Function Form a(x)b(y)c(x+y)(-1)^{q(x,y)}: The specific form of the target function is also significant. Each component plays a distinct role:
  • a(x) and b(y) capture the dependence of f on x and y separately.
  • c(x+y) captures the dependence on the sum x+y. This is particularly relevant in additive combinatorics, where sums and differences play a central role.
  • (-1)^{q(x,y)} introduces a quadratic phase. Quadratic phases are ubiquitous in Fourier analysis and signal processing. They often arise in situations where there is some underlying algebraic structure.

Potential Approaches and Challenges

So, how might one approach this conjecture? What are the potential challenges?

• Fourier Analysis: Fourier analysis over finite fields is a natural tool for tackling this problem. The Fourier transform decomposes a function into its frequency components, which can reveal hidden structures. The Gowers norm can be expressed in terms of Fourier coefficients, so it might be possible to use Fourier analysis to relate the Gowers norm condition to the correlation question.
• Additive Combinatorics Techniques: Techniques from additive combinatorics, such as the Balog-Szemerédi-Gowers theorem, could be useful. These techniques allow us to deduce structural information about sets and functions based on their additive properties.
• Algebraic Geometry: Algebraic geometry might also play a role, especially in understanding the structure of quadratic forms. Quadratic forms are well-studied objects in algebraic geometry, and their properties might shed light on the behavior of the quadratic phase factor (-1)^{q(x,y)}.

However, there are significant challenges:

• The Gowers Norm: The Gowers norm is a notoriously difficult object to work with. Despite its importance, our understanding of the Gowers norm is still incomplete. Proving results involving the Gowers norm often requires intricate arguments and sophisticated techniques.
• The Correlation Question: Establishing correlation between functions can be challenging, especially when the target function has a complex form. It might be necessary to develop new techniques for detecting correlation in this context.
• Combining Techniques: A successful approach might require combining techniques from different areas, such as Fourier analysis, additive combinatorics, and algebraic geometry. This can be difficult, as it requires expertise in multiple fields.

Why I'm Putting This Up For Grabs

While I find this conjecture incredibly interesting, my current research priorities lie elsewhere. I believe this problem is ripe for exploration, and I'd love to see someone else tackle it. It touches upon fundamental questions in mathematics and computer science, and a solution would undoubtedly be a significant contribution. So, if you're looking for a challenging and rewarding research problem, Green's Open Problem #55 might be just what you're looking for!

AMS Categories

This problem falls under the following AMS categories:

• ams-05: Combinatorics
• ams-11: Number theory

These classifications highlight the problem's connections to both discrete mathematics and the study of numbers.

Let's get this conjecture solved, people!