Weyl's Criterion

Definition

Weyl's Criterion is a fundamental result in the theory of uniform distribution that provides necessary and sufficient conditions for a sequence of real numbers to be uniformly distributed modulo 1. This criterion is closely linked to the study of the equidistribution of sequences and plays a significant role in understanding the convergence properties of sums involving these sequences.

5 Must Know Facts For Your Next Test

1. Weyl's Criterion states that a sequence $(x_n)$ is uniformly distributed modulo 1 if and only if, for every non-zero integer $k$, the average of the exponential sums $\frac{1}{N} \sum_{n=1}^N e^{2\pi i k x_n}$ converges to zero as $N \to \infty$.
2. This criterion can be applied to various sequences such as rational multiples of irrational numbers, highlighting which ones are uniformly distributed.
3. Weyl's Criterion serves as a bridge between number theory and ergodic theory, offering insights into how sequences behave under various transformations.
4. The proof of Weyl's Criterion typically employs techniques from Fourier analysis, illustrating the importance of understanding harmonic functions in this context.
5. Weyl's Criterion can also be generalized, leading to extensions in different mathematical fields including dynamic systems and probability theory.

Review Questions

• What are the conditions outlined in Weyl's Criterion for determining whether a sequence is uniformly distributed modulo 1?
  • Weyl's Criterion establishes that for a sequence $(x_n)$ to be uniformly distributed modulo 1, it must satisfy the condition that for every non-zero integer $k$, the average of the sums $\frac{1}{N} \sum_{n=1}^N e^{2\pi i k x_n}$ approaches zero as $N$ tends to infinity. This means that as we look at more terms in the sequence, their distribution should become increasingly even when considered modulo 1.
• Discuss how Weyl's Criterion can be applied to specific sequences, such as those involving irrational numbers.
  • When applying Weyl's Criterion to sequences formed by rational multiples of irrational numbers, we can determine their uniform distribution by examining how well they fill the unit interval when reduced modulo 1. For example, if we take a sequence like $(n\alpha)$ where $\\alpha$ is an irrational number, we find that this sequence is uniformly distributed modulo 1 according to Weyl's Criterion because it satisfies the required convergence condition for any integer $k$. This highlights both the power and versatility of Weyl’s Criterion in analyzing diverse sequences.
• Evaluate the implications of Weyl's Criterion in ergodic theory and its relationship with uniform distribution.
  • Weyl's Criterion has significant implications in ergodic theory, particularly concerning how dynamical systems exhibit uniform distribution properties. When applying this criterion, one can analyze invariant measures and the behavior of orbits under transformations. The criterion helps establish connections between ergodicity and uniform distribution; for instance, it aids in proving that certain sequences generated by dynamical systems are equidistributed. Understanding these relationships enriches our comprehension of both number theory and statistical properties within ergodic contexts.

Related terms

Uniform Distribution:

A sequence is said to be uniformly distributed modulo 1 if, for any interval, the proportion of terms falling within that interval approaches the length of the interval as the number of terms increases.

Equidistribution: The property of a sequence where its points become evenly spread over a certain space, often analyzed through the lens of convergence in probability or measure.

Kronecker's Theorem: A theorem stating that a sequence is uniformly distributed modulo 1 if and only if its averages converge to a constant value when integrated over any interval.