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A sequence of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l0570801.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l0570802.png" />; this class of sequences is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l0570803.png" /> and is used, in particular, in the theory of [[Lacunary series|lacunary series]] and in the theory of [[Lacunary trigonometric series|lacunary trigonometric series]]. There are generalizations of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l0570805.png" />. For example, the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l0570807.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l0570808.png" /> if there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l0570809.png" /> such that the number of solutions of the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l05708010.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l05708011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l05708012.png" /> is the integer part of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l05708013.png" />) does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l05708014.png" /> for any integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l05708015.png" />; the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l05708017.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l05708018.png" /> if there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l05708019.png" /> such that the number of solutions of the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l05708020.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l05708021.png" />) does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l05708022.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l05708023.png" /> and any integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l05708024.png" />; and the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l05708026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l05708028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l05708030.png" />, consisting of sequences that split into finitely-many sequences of the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l05708031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l05708032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057080/l05708033.png" />, respectively.
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====References====
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<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></table>
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A sequence of numbers $\seq{n_k}$ such that $n_{k+1} / n_k \geq \lambda > 1$; this class of sequences is denoted by $\Lambda$ and is used, in particular, in the theory of [[Lacunary series|lacunary series]] and in the theory of [[Lacunary trigonometric series|lacunary trigonometric series]]. There are generalizations of the class $\Lambda$. For example, the class $B_2$: $\seq{n_k} \in B_2$ if there is an $A$ such that the number of solutions of the equations $[n_{k_1} \pm n_{k_2}] = m$ (where $n_{k_1} > n_{k_2}$ and $[a]$ is the integer part of the number $a$) does not exceed $A$ for any integer $m$; the class $R$: $\seq{n_k} \in R$ if there is an $A$ such that the number of solutions of the equations $[n_{k_1} \pm \cdots \pm n_{k_p}] = m$ (where $n_{k_1} > \cdots > n_{k_p}$) does not exceed $A^p$ for any $p=2,3,\ldots$ and any integer $m$; and the classes $\Lambda_\sigma$, $B_{2\sigma}$, $R_\sigma$, consisting of sequences that split into finitely-many sequences of the classes $\Lambda$, $B_2$, $R$, respectively.
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|valign="top"|{{Ref|Ba}}||valign="top"| N.K. [N.K. Bari] Bary, "A treatise on trigonometric series", Pergamon (1964) (Translated from Russian)
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Revision as of 21:59, 3 May 2012

2010 Mathematics Subject Classification: Primary: 11Bxx [MSN][ZBL]

$ \newcommand{\seq}[1]{\left(#1\right)} % \newcommand{\seq}[1]{\left\{#1\right\}} $

A sequence of numbers $\seq{n_k}$ such that $n_{k+1} / n_k \geq \lambda > 1$; this class of sequences is denoted by $\Lambda$ and is used, in particular, in the theory of lacunary series and in the theory of lacunary trigonometric series. There are generalizations of the class $\Lambda$. For example, the class $B_2$: $\seq{n_k} \in B_2$ if there is an $A$ such that the number of solutions of the equations $[n_{k_1} \pm n_{k_2}] = m$ (where $n_{k_1} > n_{k_2}$ and $[a]$ is the integer part of the number $a$) does not exceed $A$ for any integer $m$; the class $R$: $\seq{n_k} \in R$ if there is an $A$ such that the number of solutions of the equations $[n_{k_1} \pm \cdots \pm n_{k_p}] = m$ (where $n_{k_1} > \cdots > n_{k_p}$) does not exceed $A^p$ for any $p=2,3,\ldots$ and any integer $m$; and the classes $\Lambda_\sigma$, $B_{2\sigma}$, $R_\sigma$, consisting of sequences that split into finitely-many sequences of the classes $\Lambda$, $B_2$, $R$, respectively.

References

[Ba] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series", Pergamon (1964) (Translated from Russian)
How to Cite This Entry:
Lacunary sequence. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Lacunary_sequence&oldid=25904
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