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<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(0\le q\le 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>q</mi> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\mathbb {N}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">N</mi> </math></EquationSource> </InlineEquation> denote the set of all positive integers. In this paper (which is a continuation of [<CitationRef CitationID="CR17">17</CitationRef>]) we will be interested in the family <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\({\mathcal {U}}(x^q)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">U</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mi>q</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of all regularly distributed sets <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(X \subset {\mathbb {N}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>⊂</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> whose ratio block sequence of type (<InternalRef RefID="Equ1">1.1</InternalRef>) is asymptotically distributed with distribution function <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(g(x) = x^q;\ x \in [0,1]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>x</mi> <mi>q</mi> </msup> <mo>;</mo> <mspace width="4pt" /> <mi>x</mi> <mo>∈</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We will study the structure of these families with respect to the union, intersection and difference of the sets <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(X\in {\mathcal {U}}(x^p)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>∈</mo> <mi mathvariant="script">U</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mi>p</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(Y\in {\mathcal {U}}(x^q)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Y</mi> <mo>∈</mo> <mi mathvariant="script">U</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mi>q</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(0\le p, q\le 1.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>≤</mo> <mn>1</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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On structure of the family of regularly distributed sets

  • 期刊论文

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  • 发表日期：

    2025年11月27日

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  • János T. Tóth,
  • Ferdinánd Filip,
  • Piotr Miska,
  • József Bukor

摘要

Let \(0\le q\le 1\) 0 ≤ q ≤ 1 and \({\mathbb {N}}\) N denote the set of all positive integers. In this paper (which is a continuation of [17]) we will be interested in the family \({\mathcal {U}}(x^q)\) U ( x q ) of all regularly distributed sets \(X \subset {\mathbb {N}}\) X ⊂ N whose ratio block sequence of type (1.1) is asymptotically distributed with distribution function \(g(x) = x^q;\ x \in [0,1]\) g ( x ) = x q ; x ∈ [ 0 , 1 ] . We will study the structure of these families with respect to the union, intersection and difference of the sets \(X\in {\mathcal {U}}(x^p)\) X ∈ U ( x p ) , \(Y\in {\mathcal {U}}(x^q)\) Y ∈ U ( x q ) , where \(0\le p, q\le 1.\) 0 ≤ p , q ≤ 1 .