# Uniqueness set

$U$- set
A set $E \subset [ 0 ,\ 2 \pi ]$ such that a trigonometric series that converges to zero at each point of $( 0 ,\ 2 \pi ] \setminus E$ is the zero series. A set that is not a $U$- set is a called a set of non-uniqueness, or an $M$- set. These concepts are related to the problem of the uniqueness of the representation of a function by a trigonometric series converging to it everywhere, except perhaps on a given set $E$. G. Cantor (1872) showed that a finite set (including the empty set) is a set of uniqueness, and the extension of this result to infinite sets led him to the creation of set theory.
Sets of positive Lebesgue measure are always $M$- sets. Any countable set is a $U$- set. There exists perfect sets (cf. Perfect set) of measure zero that are $M$- sets (D.E. Men'shov, 1916), and ones that are $U$- sets (N.K. Bari, 1921); for example, the Cantor set with a constant rational ratio $\theta$ is a $U$- set if and only $1 / \theta$ is an integer, that is, whether a set of numbers is a $U$- set or an $M$- set depends on the arithmetical nature of the numbers forming it. However, there exist sets $E \subset [ 0 ,\ 2 \pi ]$ of full measure (so-called $U ( \epsilon )$- sets) such that any trigonometric series that converges to zero at every point of $[ 0 ,\ 2 \pi ] \setminus E$ and has coefficients that are $O ( \epsilon _{n} )$, where $\epsilon _{n} \downarrow 0$, is the zero series.
The concepts of $U$- sets and $M$- sets can be generalized to Fourier–Stieltjes series.