**Weyl Sum**

The Weyl sum, an object of central importance in analytic number theory, is defined as follows:

$$S(N, \alpha) = \sum_{n=1}^{N} e^{2\pi i f(n) \alpha}$$

where $N$ is a positive integer and $\alpha$ is a real number. The behavior of this sum as $N$ grows large is of particular interest. The size of the Weyl sum is often related to the behavior of $f(n)$. Generally, if we impose some regularity on $f(n)$, such as continuous derivatives, Lipschitz continuity, or continuous second derivatives, we can control the exponential sum to some extent.

The most classical method to control the Weyl sum is the so-called Weyl differencing technique, which reduces the complexity of $f$ by transferring this complexity to the norm.

Vinogradov’s method cleverly uses the regularity of $f$ to minimize the losses incurred during this complexity transfer.

**Ergodicity of the Shift Map**

The shift map on $T$ is a fundamental example of a dynamical system. It is defined by the transformation:

$$T: x \mapsto x + \alpha \mod 1$$

We shall use the following terminology. Let $\mathbb{R} / \mathbb{Z}$ be identified with the interval $[0,1]/\sim$ (with addition modulo 1 ) and let $\lambda$ denote the normalized Haar measure on $\mathbb{R} / \mathbb{Z} \cong[0,1]$. If $n \in \mathbb{N}$ and $x \in[0,1]$, we write

$$

\varphi_n(x):=\varphi(x)+\varphi({x+\alpha})+\cdots+\varphi({x+(n-1) \alpha}),

$$

where ${\cdot}$ denotes the fractional part.

A key point to grasp is whether $\phi(0)=\phi(1)$, as this essentially reflects whether the rotation degree of the system implies an average energy of zero, indicating no rotation occurs. If $\phi(0)\neq \phi(1)$, it suggests that the energy of the dynamical system changes over time, indicating a spatial gauge variation across the circle.

It is important to note that the two scenarios are different. If $\phi(0)=\phi(1)$, the energy remains constant throughout the dynamical system’s evolution. However, if $\phi(0)\neq \phi(1)$, there will be an energy change, which is a gauge variation in the fiber bundle.

These considerations naturally lead to questions about the dynamics of the system: whether it is stationary or not, whether a point will diffuse or conserve energy, It is well-known that the following two questions are closely related to the ergodicity of skew products of the type $T_{\phi}$ and what can be said about the limit points of the system, which are crucial for understanding the dynamics.

These two questions are of interest in the theory of uniform distribution modulo 1 as well.

**Connection Between Weyl Sum and Ergodicity**

Regarding the relationship between this problem and the Weyl sum, in the end, you can reduce it to a problem of bounding the Weyl sum. However, this reduction itself is not so trivial and involves some calculations, but at once you are able to bound the Weyl sum, the non-trivial bound of the Weyl sum will at least provide some results for these two problems.