---
tag: K7CNWXF9SN
kind: concept
name: Set of uniqueness
msc: [42A63]
sources:
  - url: https://en.wikipedia.org/wiki/Set_of_uniqueness
same_as:
  wikidata: Q7456288
  wikipedia: Set of uniqueness
---

## Summary

A set of uniqueness is a subset $E$ of the circle with the following property.
A [[trigonometric series|FZK2M8KGXQ]] that converges to $0$ at every point outside $E$ must have all coefficients equal to $0$.
A set without this property is a set of multiplicity.
[[Cantor's uniqueness theorem|NSS3XW4BYZ]] says that the empty set is a set of uniqueness.
The question of which exceptional sets are allowed led Cantor to his theory of sets and ordinals.
The study of these sets connects harmonic analysis with measure theory, number theory and descriptive set theory.

## Explanations

### L1 human

Suppose a sum of waves is known to be $0$ everywhere except on a set $E$, where nothing is known.
If $E$ is a set of uniqueness, this already forces every wave in the sum to vanish.
Small sets, such as finite or countable sets, are sets of uniqueness.
Large sets, such as intervals, are not.
Surprisingly, Menshov found a closed set of length zero that is not a set of uniqueness, so being small in length is not enough.