Scholarlink

<p>Consider a set <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(X\subseteq {\mathbb {R}}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>⊆</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> which is 1-dense, namely, it intersects every open unit ball. We show that for any <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n\in {\mathbb {N}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, we can get from any point to any other point in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\({\mathbb {R}}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> in <i>n</i> steps so that the intermediate points are in <i>X</i>, and the discrepancy of the step vectors is at most <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(2\sqrt{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <msqrt> <mn>2</mn> </msqrt> </mrow> </math></EquationSource> </InlineEquation>.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Block Partitions in Higher Dimensions

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2025年4月3日

查看全文
  • Endre Csóka

摘要

Consider a set \(X\subseteq {\mathbb {R}}^d\) X ⊆ R d which is 1-dense, namely, it intersects every open unit ball. We show that for any \(n\in {\mathbb {N}}\) n ∈ N , we can get from any point to any other point in \({\mathbb {R}}^d\) R d in n steps so that the intermediate points are in X, and the discrepancy of the step vectors is at most \(2\sqrt{2}\) 2 2 .