Weyl's inequality (number theory)

Weyl's Inequality is a fundamental result in analytic number theory that provides a bound on the exponential sum

$\left| \sum_{n=1}^N e^{2\pi i f(n)} \right|$

where $f(x)$ is a real-valued polynomial and $N$ is a positive integer. It allows one to show that if a polynomial has sufficient rational approximations to its coefficients, then the exponential sum is relatively small.

Statement:

Let $f(x) = \alpha_k x^k + \alpha_{k-1} x^{k-1} + \cdots + \alpha_1 x + \alpha_0$ be a real polynomial of degree $k \geq 2$. Suppose that there exist integers $a$ and $q$ such that $q > 0$, $(a, q) = 1$, and

$\left| \alpha_k - \frac{a}{q} \right| \leq \frac{1}{q^2}$.

Then, for any $\epsilon > 0$,

$\left| \sum_{n=1}^N e^{2\pi i f(n)} \right| \ll N^{1 + \epsilon} \left( \frac{1}{q} + \frac{1}{N} + \frac{q}{N^k} \right)^{1/(2^{k-1})}$,

where the implied constant depends only on $k$ and $\epsilon$. The notation $\ll$ means "less than or equal to up to a constant factor."

Significance and Applications:

Weyl's inequality is crucial for studying the distribution of sequences modulo 1, particularly for polynomial sequences. It has numerous applications, including:

• Waring's Problem: Estimating the number of solutions to equations of the form $n = x_1^k + \cdots + x_s^k$, where $n$ is a given integer and $x_1, \dots, x_s$ are non-negative integers. Weyl's inequality provides bounds on the number of integers required to express any sufficiently large integer as a sum of $k$th powers.

• Distribution of Polynomial Sequences Modulo 1: Determining how uniformly distributed the sequence ${f(n)}$ is modulo 1. If the exponential sum is small, the sequence is "well-distributed".

• Zeros of the Riemann Zeta Function: Although not a direct application, Weyl's inequality is related to techniques used in zero-density estimates for the Riemann zeta function.

History:

Weyl's inequality was developed by Hermann Weyl in the early 20th century. It significantly improved upon earlier results and laid the foundation for many subsequent developments in analytic number theory.

Generalizations and Related Results:

• Vinogradov's Mean Value Theorem: This provides a stronger estimate for exponential sums with polynomial phases than Weyl's inequality, although it is more complicated to prove.

• Vaughan's Identity: A powerful tool used in conjunction with Weyl's inequality and other techniques to address problems in additive number theory.

Limitations:

Weyl's inequality, while powerful, has limitations. The exponent $1/(2^{k-1})$ becomes very small as the degree $k$ increases, making the estimate less effective for polynomials of high degree. Vinogradov's mean value theorem provides a better estimate in these cases.