# An introduction to Vinogradov’s mean value theorem

VMT-Intro: pdf file containing the post

A link to a talk I gave on the topic. The talk is much more elementary than the blog post.

Vinogradov’s mean value theorem (recently proved by Bourgain, Demeter and Guth) is a beautiful problem in the intersection of number theory and harmonic analysis.

As a problem in number theory, the mean value theorem requests to show most of the solutions to a certain system of Diophantine equations of the form $f_j(\vec{x}) = f_j(\vec{y})$ for some integral polynomials $f_1 , \ldots , f_k$ are of the form $\vec{x} = \vec{y}$. As a problem in harmonic analysis, the mean value theorem requests for an upper bound for the $L^p$ norm of a certain function. These two problems turn out to be equivalent, as explained in the pdf linked at the top of the post, thanks to Fourier analytic identities.

One goal in the above pdf is to understand the nature of the so called “critical exponent.” Interpolation reveals that Vinogradov’s mean value theorem follows from an $L^{k(k+1)}$ bound of a certain function. While a first step to understanding this critical exponent is interpolation, consideration of the major arcs gives proper insight into why $k(k+1)$ appears.

In the final section, I attempt to explain how the mean value theorem can be interpreted as a stronger form of orthogonality of certain complex exponentials. For a vector $n \in \mathbb{Z}^k$, we define $e_n : \mathbb{R}^k / \mathbb{Z}^k \to \mathbb{C}$ via $e_n(\alpha) : = e^{2 \pi i n \cdot \alpha}$. Then Vinogradov’s mean value theorem can be interpreted as showing $\{e_{(j , \ldots , j^k)}\}_{1 \leq j \leq N}$ are stronger than orthogonal ($k \geq 2$, not true for $k =1$). We make this somewhat precise in the above pdf, adopting a probabilistic perspective.

I’d like to thank Chris Gartland, a fellow graduate student, for helping me formulate the ideas in the last section. For instance, it was his idea to utilize equation 5.

I’d also like to thank Junxian Li, a fellow graduate student in number theory, for very useful discussions regarding the major arcs.

Lastly, anyone at UIUC interested in hearing more about Vinogradov’s mean value theorem (for instance Bourgain, Demeter and Guth’s recent result or classical number theoretic methods), please get in touch with me. My email can be found here, or you can visit my office in Altgeld 169.