View source for Uniqueness set

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u0954402.png" />-set''

A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u0954403.png" /> such that a [[Trigonometric series|trigonometric series]] that converges to zero at each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u0954404.png" /> is the zero series. A set that is not a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u0954405.png" />-set is a called a set of non-uniqueness, or an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u0954407.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u0954408.png" />. 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|set theory]].

Sets of positive [[Lebesgue measure|Lebesgue measure]] are always <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u0954409.png" />-sets. Any countable set is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544010.png" />-set. There exists perfect sets (cf. [[Perfect set|Perfect set]]) of measure zero that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544011.png" />-sets (D.E. Men'shov, 1916), and ones that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544012.png" />-sets (N.K. Bari, 1921); for example, the [[Cantor set|Cantor set]] with a constant rational ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544013.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544014.png" />-set if and only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544015.png" /> is an integer, that is, whether a set of numbers is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544016.png" />-set or an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544017.png" />-set depends on the arithmetical nature of the numbers forming it. However, there exist sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544018.png" /> of full measure (so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544020.png" />-sets) such that any trigonometric series that converges to zero at every point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544021.png" /> and has coefficients that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544023.png" />, is the zero series.

The concepts of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544024.png" />-sets and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544025.png" />-sets can be generalized to Fourier–Stieltjes series.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,   "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR></table>



====Comments====
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544026.png" />-sets are also called sets of multiplicity. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544027.png" /> such that a [[Fourier–Stieltjes series|Fourier–Stieltjes series]] that converges to zero at each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544028.png" /> is the zero series, is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544030.png" />-set, or a set of extended uniqueness. A set that is not a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544031.png" />-set is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544033.png" />-set, or a set of restricted multiplicity. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544034.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544035.png" />-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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544036.png" />-sets play a more prominent role than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544037.png" />-sets. In 1983, R. Lyons proved that the Rajchman measures are exactly the measures that annihilate all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544038.png" />-sets. In [[#References|[a1]]]–[[#References|[a3]]] many more results are given, e.g. relating uniqueness sets with Helson sets and sets of spectral synthesis (cf. [[Harmonic analysis, abstract|Harmonic analysis, abstract]]).

Consider a closed interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544039.png" /> of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544040.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544042.png" />, be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544043.png" /> numbers and consider the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544044.png" /> closed intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544045.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544046.png" /> 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544048.png" /></td> </tr></table>

Now start with any interval of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544049.png" />. Perform a dissection of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544050.png" />, perform a dissection of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544051.png" /> on each of the intervals obtained, etc. After <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544052.png" /> iterations one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544053.png" /> intervals, each of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544054.png" />, and as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544055.png" /> the final result is a closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544056.png" /> of measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544057.png" /> (the limit exists). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544058.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544059.png" />, the resulting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544060.png" /> is perfect (cf. [[Perfect set|Perfect set]]) and non-dense. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544064.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544065.png" />, one obtains the [[Cantor set|Cantor set]]. Taking successive dissections of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544066.png" /> yields a so-called set of Cantor type. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544067.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544068.png" />, one speaks of a set of Cantor type of constant ratio (of dissection). Cf. [[#References|[2]]], pp. 194ff, for more details.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.C. Graham,   O.C. McGehee,   "Essays in commutative harmonic analysis" , Springer  (1979)  pp. Chapt. 5</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.-P. Kahane,   "Séries de Fourier absolument convergentes" , Springer  (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.S. Kechris,   A. Louveau,   "Descriptive set theory and the structure of sets of uniqueness" , Cambridge Univ. Press  (1987)</TD></TR></table>