# 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.

#### References

 [1] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian) [2] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) [3] N.K. Bari, "The uniqueness problem of the representation of functions by trigonometric series" Transl. Amer. Math. Soc. (1) , 3 (1951) pp. 107–195 Uspekhi Mat. Nauk , 4 : 3 (1949) pp. 3–68

$M$- sets are also called sets of multiplicity. A set $E \subset [ 0,\ 2 \pi ]$ such that a Fourier–Stieltjes series that converges to zero at each point of $( 0,\ 2 \pi ] \setminus E$ is the zero series, is called a $U _{0}$- set, or a set of extended uniqueness. A set that is not a $U _{0}$- set is called an $M _{0}$- set, or a set of restricted multiplicity. A set $E$ is a $U _{0}$- set if and only if it does not support a non-zero Rajchman measure, that is, a measure whose Fourier–Stieltjes coefficients tend to zero at infinity. In the modern theory, $U _{0}$- sets play a more prominent role than $U$- sets. In 1983, R. Lyons proved that the Rajchman measures are exactly the measures that annihilate all $U _{0}$- sets. In [a1][a3] many more results are given, e.g. relating uniqueness sets with Helson sets and sets of spectral synthesis (cf. Harmonic analysis, abstract).

Consider a closed interval $[ x,\ x+l ]$ of length $l$. Let $\alpha (1) \dots \alpha (d)$, $0 \leq \alpha (1) < \alpha (2) < \dots < \alpha (d) < 1$, be $d$ numbers and consider the $d$ closed intervals $[ x+ \alpha (j) l,\ x + \alpha (j)l + \eta ]$, where $\eta$ is small enough so that the intervals have no points in common. Retain only these intervals (and throw the complementary intervals away). This is referred to as performing a dissection of type

$$[ d ; \ \alpha (1) \dots \alpha (d) ; \ \eta ] .$$

Now start with any interval of length $m$. Perform a dissection of type $[ d _{1} ; \ \alpha _{1} (1) \dots \alpha _{1} (d _{1} ) ; \ \eta _{1} ]$, perform a dissection of type $[ d _{2} ; \ \alpha _{2} (1) \dots \alpha _{2} (d _{2} ) ; \ \eta _{2} ]$ on each of the intervals obtained, etc. After $p$ iterations one has $d _{1} \dots d _{p}$ intervals, each of length $\eta _{1} \dots \eta _{p} m$, and as $p \rightarrow \infty$ the final result is a closed set $P$ of measure $m \ \lim\limits _{p} \ d _{1} \dots d _{p} \eta _{1} \dots \eta _{p}$( the limit exists). If $d _{p} \geq 2$ for all $p$, the resulting $P$ is perfect (cf. Perfect set) and non-dense. For $d _{p} = 2$ and $\alpha _{p} (1) = 0$, $\alpha _{p} (2) = 2/3$, $\eta _{p} = 1/3$ for all $p$, one obtains the Cantor set. Taking successive dissections of type $[ 2 ; \ 0,\ 1 - \xi _{k} ; \ \xi _{k} ]$ yields a so-called set of Cantor type. If $\xi _{k} = \xi$ for all $k$, one speaks of a set of Cantor type of constant ratio (of dissection). Cf. [2], pp. 194ff, for more details.

#### References

 [a1] C.C. Graham, O.C. McGehee, "Essays in commutative harmonic analysis" , Springer (1979) pp. Chapt. 5 [a2] J.-P. Kahane, "Séries de Fourier absolument convergentes" , Springer (1970) [a3] A.S. Kechris, A. Louveau, "Descriptive set theory and the structure of sets of uniqueness" , Cambridge Univ. Press (1987)
How to Cite This Entry:
Uniqueness set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniqueness_set&oldid=44370
This article was adapted from an original article by V.F. Emel'yanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article