Zeta functions of discrete self-similar sets

被引：8

作者：

Essouabri, Driss ^{[1
]}

Lichtin, Ben ^{[1
]}

机构：

[1] Univ St Etienne, PRES Univ Lyon, Fac Sci & Tech, Inst Camille Jordan,CNRS,UMR 5208,Dept Math, F-42023 St Etienne 2, France

来源：

ADVANCES IN MATHEMATICS | 2013年 / 232卷 / 01期

关键词：

Self-similar discrete sets; Fractals; Zeta functions; Meromorphic continuation; Hausdorff dimension; Erdos distance problem; Ar'nold sail; CONTINUED FRACTIONS; DIMENSION; FRACTALS;

D O I：

10.1016/j.aim.2012.09.011

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

In this paper we study a class of countable and discrete subsets of a Euclidean space that are "self-similar" with respect to a finite set of (affine) similarities. Any such set can be interpreted as having a fractal structure. We introduce a zeta function for these sets, and derive basic analytic properties of this "fractal" zeta function. Motivating examples that come from combinatorial geometry and arithmetic are given particular attention. (C) 2012 Elsevier Inc. All rights reserved.

引用

页码：142 / 187

页数：46

共 33 条

[1] Dimensions of fractals in the large [J].

Allen, M. R. ;

Cruttwell, G. S. H. ;

Hare, Kathryn E. ;

Ronning, J. -O. .

CHAOS SOLITONS & FRACTALS, 2007, 31 (01) :5-13

[2]

[Anonymous], 1932, The Distribution of Prime Numbers

[3]

[Anonymous], JAPAN J MATH

[4] A-GRADED ALGEBRAS AND CONTINUED FRACTIONS [J].

ARNOLD, VI .

COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 1989, 42 (07) :993-1000

[5]

ARUTYUNYANTS G, 2004, CONT MATH, V342, P15

[6] Distribution of multinomial and q-binomial coefficients modulo p [J].

Barbolosi, D ;

Grabner, PJ .

INDAGATIONES MATHEMATICAE-NEW SERIES, 1996, 7 (02) :129-135

[7]

BARLOW MT, 1992, P LOND MATH SOC, V64, P125

[8] FRACTIONAL DIMENSION OF SETS IN DISCRETE SPACES [J].

BARLOW, MT ;

TAYLOR, SJ .

JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, 1989, 22 (13) :2621-2626

[9]

Borevich Z.I., 1966, Number Theory

[10]

Delange H., 1954, ANN SCI ECOLE NORM S, V71, P213

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