Characteristics of Distributions of Sets and Their (R)- and (N)-Denseness

Abstract

Let 0le qle 1 and mathbb {N} denotes the set of all positive integers. In this paper we will deal with it too the family {mathcal {U}}(x^q) of all regularly distributed set X={x_1<x_2<cdots<x_n<cdots } subset mathbb {N} whose ratio block sequence x1x1,x1x2,x2x2,x1x3,x2x3,x3x3,⋯,x1xn,x2xn,⋯,xnxn,⋯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{x_1}{x_1}, \\frac{x_1}{x_2}, \\frac{x_2}{x_2}, \\frac{x_1}{x_3}, \\frac{x_2}{x_3}, \\frac{x_3}{x_3}, \\dots , \\frac{x_1}{x_n}, \\frac{x_2}{x_n}, \\dots , \\frac{x_n}{x_n}, \\dots \\end{aligned}$$\\end{document}is asymptotically distributed with distribution function g(x) = x^q; x in (0,1], and we will show that the regular distributed set, regular sequences, regular variation at infinity are equivalent notations. In this paper also we discuss the relationship between notations as (N)-denseness, directions sets, generalized ratio sets, dispersion and exponent of convergence.