{"title":"重型线路的覆盖范围有多大?","authors":"Damian Dąbrowski, Tuomas Orponen, Hong Wang","doi":"10.1112/jlms.12910","DOIUrl":null,"url":null,"abstract":"<p>One formulation of Marstrand's slicing theorem is the following. Assume that <span></span><math>\n <semantics>\n <mrow>\n <mi>t</mi>\n <mo>∈</mo>\n <mo>(</mo>\n <mn>1</mn>\n <mo>,</mo>\n <mn>2</mn>\n <mo>]</mo>\n </mrow>\n <annotation>$t \\in (1,2]$</annotation>\n </semantics></math>, and <span></span><math>\n <semantics>\n <mrow>\n <mi>B</mi>\n <mo>⊂</mo>\n <msup>\n <mi>R</mi>\n <mn>2</mn>\n </msup>\n </mrow>\n <annotation>$B \\subset \\mathbb {R}^{2}$</annotation>\n </semantics></math> is a Borel set with <span></span><math>\n <semantics>\n <mrow>\n <msup>\n <mi>H</mi>\n <mi>t</mi>\n </msup>\n <mrow>\n <mo>(</mo>\n <mi>B</mi>\n <mo>)</mo>\n </mrow>\n <mo><</mo>\n <mi>∞</mi>\n </mrow>\n <annotation>$\\mathcal {H}^{t}(B) &lt; \\infty$</annotation>\n </semantics></math>. Then, for almost all directions <span></span><math>\n <semantics>\n <mrow>\n <mi>e</mi>\n <mo>∈</mo>\n <msup>\n <mi>S</mi>\n <mn>1</mn>\n </msup>\n </mrow>\n <annotation>$e \\in S^{1}$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <msup>\n <mi>H</mi>\n <mi>t</mi>\n </msup>\n <annotation>$\\mathcal {H}^{t}$</annotation>\n </semantics></math> almost all of <span></span><math>\n <semantics>\n <mi>B</mi>\n <annotation>$B$</annotation>\n </semantics></math> is covered by lines <span></span><math>\n <semantics>\n <mi>ℓ</mi>\n <annotation>$\\ell$</annotation>\n </semantics></math> parallel to <span></span><math>\n <semantics>\n <mi>e</mi>\n <annotation>$e$</annotation>\n </semantics></math> with <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mo>dim</mo>\n <mi>H</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>B</mi>\n <mo>∩</mo>\n <mi>ℓ</mi>\n <mo>)</mo>\n </mrow>\n <mo>=</mo>\n <mi>t</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$\\dim _{\\mathrm{H}}(B \\cap \\ell) = t - 1$</annotation>\n </semantics></math>. We investigate the prospects of sharpening Marstrand's result in the following sense: in a generic direction <span></span><math>\n <semantics>\n <mrow>\n <mi>e</mi>\n <mo>∈</mo>\n <msup>\n <mi>S</mi>\n <mn>1</mn>\n </msup>\n </mrow>\n <annotation>$e \\in S^{1}$</annotation>\n </semantics></math>, is it true that a strictly less than <span></span><math>\n <semantics>\n <mi>t</mi>\n <annotation>$t$</annotation>\n </semantics></math>-dimensional part of <span></span><math>\n <semantics>\n <mi>B</mi>\n <annotation>$B$</annotation>\n </semantics></math> is covered by the heavy lines <span></span><math>\n <semantics>\n <mrow>\n <mi>ℓ</mi>\n <mo>⊂</mo>\n <msup>\n <mi>R</mi>\n <mn>2</mn>\n </msup>\n </mrow>\n <annotation>$\\ell \\subset \\mathbb {R}^{2}$</annotation>\n </semantics></math>, namely those with <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mo>dim</mo>\n <mi>H</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>B</mi>\n <mo>∩</mo>\n <mi>ℓ</mi>\n <mo>)</mo>\n </mrow>\n <mo>></mo>\n <mi>t</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$\\dim _{\\mathrm{H}} (B \\cap \\ell) &gt; t - 1$</annotation>\n </semantics></math>? A positive answer for <span></span><math>\n <semantics>\n <mi>t</mi>\n <annotation>$t$</annotation>\n </semantics></math>-regular sets <span></span><math>\n <semantics>\n <mrow>\n <mi>B</mi>\n <mo>⊂</mo>\n <msup>\n <mi>R</mi>\n <mn>2</mn>\n </msup>\n </mrow>\n <annotation>$B \\subset \\mathbb {R}^{2}$</annotation>\n </semantics></math> was previously obtained by the first author. The answer for general Borel sets turns out to be negative for <span></span><math>\n <semantics>\n <mrow>\n <mi>t</mi>\n <mo>∈</mo>\n <mo>(</mo>\n <mn>1</mn>\n <mo>,</mo>\n <mstyle>\n <mfrac>\n <mn>3</mn>\n <mn>2</mn>\n </mfrac>\n </mstyle>\n <mo>]</mo>\n </mrow>\n <annotation>$t \\in (1,\\tfrac{3}{2}]$</annotation>\n </semantics></math> and positive for <span></span><math>\n <semantics>\n <mrow>\n <mi>t</mi>\n <mo>∈</mo>\n <mo>(</mo>\n <mstyle>\n <mfrac>\n <mn>3</mn>\n <mn>2</mn>\n </mfrac>\n </mstyle>\n <mo>,</mo>\n <mn>2</mn>\n <mo>]</mo>\n </mrow>\n <annotation>$t \\in (\\tfrac{3}{2},2]$</annotation>\n </semantics></math>. More precisely, the heavy lines can cover up to a <span></span><math>\n <semantics>\n <mrow>\n <mi>min</mi>\n <mo>{</mo>\n <mi>t</mi>\n <mo>,</mo>\n <mn>3</mn>\n <mo>−</mo>\n <mi>t</mi>\n <mo>}</mo>\n </mrow>\n <annotation>$\\min \\lbrace t,3 - t\\rbrace$</annotation>\n </semantics></math> dimensional part of <span></span><math>\n <semantics>\n <mi>B</mi>\n <annotation>$B$</annotation>\n </semantics></math> in a generic direction. We also consider the part of <span></span><math>\n <semantics>\n <mi>B</mi>\n <annotation>$B$</annotation>\n </semantics></math> covered by the <span></span><math>\n <semantics>\n <mi>s</mi>\n <annotation>$s$</annotation>\n </semantics></math>-heavy lines, namely those with <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mo>dim</mo>\n <mi>H</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>B</mi>\n <mo>∩</mo>\n <mi>ℓ</mi>\n <mo>)</mo>\n </mrow>\n <mo>⩾</mo>\n <mi>s</mi>\n </mrow>\n <annotation>$\\dim _{\\mathrm{H}}(B \\cap \\ell) \\geqslant s$</annotation>\n </semantics></math> for <span></span><math>\n <semantics>\n <mrow>\n <mi>s</mi>\n <mo>></mo>\n <mi>t</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$s &gt; t - 1$</annotation>\n </semantics></math>. We establish a sharp answer to the question: how much can the <span></span><math>\n <semantics>\n <mi>s</mi>\n <annotation>$s$</annotation>\n </semantics></math>-heavy lines cover in a generic direction? Finally, we identify a new class of sets called sub-uniformly distributed sets, which generalise Ahlfors-regular sets. Roughly speaking, these sets share the spatial uniformity of Ahlfors-regular sets, but pose no restrictions on uniformity across different scales. We then extend and sharpen the first author's previous result on Ahlfors-regular sets to the class of sub-uniformly distributed sets.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12910","citationCount":"0","resultStr":"{\"title\":\"How much can heavy lines cover?\",\"authors\":\"Damian Dąbrowski, Tuomas Orponen, Hong Wang\",\"doi\":\"10.1112/jlms.12910\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>One formulation of Marstrand's slicing theorem is the following. Assume that <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>t</mi>\\n <mo>∈</mo>\\n <mo>(</mo>\\n <mn>1</mn>\\n <mo>,</mo>\\n <mn>2</mn>\\n <mo>]</mo>\\n </mrow>\\n <annotation>$t \\\\in (1,2]$</annotation>\\n </semantics></math>, and <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>B</mi>\\n <mo>⊂</mo>\\n <msup>\\n <mi>R</mi>\\n <mn>2</mn>\\n </msup>\\n </mrow>\\n <annotation>$B \\\\subset \\\\mathbb {R}^{2}$</annotation>\\n </semantics></math> is a Borel set with <span></span><math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mi>H</mi>\\n <mi>t</mi>\\n </msup>\\n <mrow>\\n <mo>(</mo>\\n <mi>B</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo><</mo>\\n <mi>∞</mi>\\n </mrow>\\n <annotation>$\\\\mathcal {H}^{t}(B) &lt; \\\\infty$</annotation>\\n </semantics></math>. Then, for almost all directions <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>e</mi>\\n <mo>∈</mo>\\n <msup>\\n <mi>S</mi>\\n <mn>1</mn>\\n </msup>\\n </mrow>\\n <annotation>$e \\\\in S^{1}$</annotation>\\n </semantics></math>, <span></span><math>\\n <semantics>\\n <msup>\\n <mi>H</mi>\\n <mi>t</mi>\\n </msup>\\n <annotation>$\\\\mathcal {H}^{t}$</annotation>\\n </semantics></math> almost all of <span></span><math>\\n <semantics>\\n <mi>B</mi>\\n <annotation>$B$</annotation>\\n </semantics></math> is covered by lines <span></span><math>\\n <semantics>\\n <mi>ℓ</mi>\\n <annotation>$\\\\ell$</annotation>\\n </semantics></math> parallel to <span></span><math>\\n <semantics>\\n <mi>e</mi>\\n <annotation>$e$</annotation>\\n </semantics></math> with <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mo>dim</mo>\\n <mi>H</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>B</mi>\\n <mo>∩</mo>\\n <mi>ℓ</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>=</mo>\\n <mi>t</mi>\\n <mo>−</mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$\\\\dim _{\\\\mathrm{H}}(B \\\\cap \\\\ell) = t - 1$</annotation>\\n </semantics></math>. We investigate the prospects of sharpening Marstrand's result in the following sense: in a generic direction <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>e</mi>\\n <mo>∈</mo>\\n <msup>\\n <mi>S</mi>\\n <mn>1</mn>\\n </msup>\\n </mrow>\\n <annotation>$e \\\\in S^{1}$</annotation>\\n </semantics></math>, is it true that a strictly less than <span></span><math>\\n <semantics>\\n <mi>t</mi>\\n <annotation>$t$</annotation>\\n </semantics></math>-dimensional part of <span></span><math>\\n <semantics>\\n <mi>B</mi>\\n <annotation>$B$</annotation>\\n </semantics></math> is covered by the heavy lines <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>ℓ</mi>\\n <mo>⊂</mo>\\n <msup>\\n <mi>R</mi>\\n <mn>2</mn>\\n </msup>\\n </mrow>\\n <annotation>$\\\\ell \\\\subset \\\\mathbb {R}^{2}$</annotation>\\n </semantics></math>, namely those with <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mo>dim</mo>\\n <mi>H</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>B</mi>\\n <mo>∩</mo>\\n <mi>ℓ</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>></mo>\\n <mi>t</mi>\\n <mo>−</mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$\\\\dim _{\\\\mathrm{H}} (B \\\\cap \\\\ell) &gt; t - 1$</annotation>\\n </semantics></math>? A positive answer for <span></span><math>\\n <semantics>\\n <mi>t</mi>\\n <annotation>$t$</annotation>\\n </semantics></math>-regular sets <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>B</mi>\\n <mo>⊂</mo>\\n <msup>\\n <mi>R</mi>\\n <mn>2</mn>\\n </msup>\\n </mrow>\\n <annotation>$B \\\\subset \\\\mathbb {R}^{2}$</annotation>\\n </semantics></math> was previously obtained by the first author. The answer for general Borel sets turns out to be negative for <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>t</mi>\\n <mo>∈</mo>\\n <mo>(</mo>\\n <mn>1</mn>\\n <mo>,</mo>\\n <mstyle>\\n <mfrac>\\n <mn>3</mn>\\n <mn>2</mn>\\n </mfrac>\\n </mstyle>\\n <mo>]</mo>\\n </mrow>\\n <annotation>$t \\\\in (1,\\\\tfrac{3}{2}]$</annotation>\\n </semantics></math> and positive for <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>t</mi>\\n <mo>∈</mo>\\n <mo>(</mo>\\n <mstyle>\\n <mfrac>\\n <mn>3</mn>\\n <mn>2</mn>\\n </mfrac>\\n </mstyle>\\n <mo>,</mo>\\n <mn>2</mn>\\n <mo>]</mo>\\n </mrow>\\n <annotation>$t \\\\in (\\\\tfrac{3}{2},2]$</annotation>\\n </semantics></math>. More precisely, the heavy lines can cover up to a <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>min</mi>\\n <mo>{</mo>\\n <mi>t</mi>\\n <mo>,</mo>\\n <mn>3</mn>\\n <mo>−</mo>\\n <mi>t</mi>\\n <mo>}</mo>\\n </mrow>\\n <annotation>$\\\\min \\\\lbrace t,3 - t\\\\rbrace$</annotation>\\n </semantics></math> dimensional part of <span></span><math>\\n <semantics>\\n <mi>B</mi>\\n <annotation>$B$</annotation>\\n </semantics></math> in a generic direction. We also consider the part of <span></span><math>\\n <semantics>\\n <mi>B</mi>\\n <annotation>$B$</annotation>\\n </semantics></math> covered by the <span></span><math>\\n <semantics>\\n <mi>s</mi>\\n <annotation>$s$</annotation>\\n </semantics></math>-heavy lines, namely those with <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mo>dim</mo>\\n <mi>H</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>B</mi>\\n <mo>∩</mo>\\n <mi>ℓ</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>⩾</mo>\\n <mi>s</mi>\\n </mrow>\\n <annotation>$\\\\dim _{\\\\mathrm{H}}(B \\\\cap \\\\ell) \\\\geqslant s$</annotation>\\n </semantics></math> for <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>s</mi>\\n <mo>></mo>\\n <mi>t</mi>\\n <mo>−</mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$s &gt; t - 1$</annotation>\\n </semantics></math>. We establish a sharp answer to the question: how much can the <span></span><math>\\n <semantics>\\n <mi>s</mi>\\n <annotation>$s$</annotation>\\n </semantics></math>-heavy lines cover in a generic direction? Finally, we identify a new class of sets called sub-uniformly distributed sets, which generalise Ahlfors-regular sets. Roughly speaking, these sets share the spatial uniformity of Ahlfors-regular sets, but pose no restrictions on uniformity across different scales. We then extend and sharpen the first author's previous result on Ahlfors-regular sets to the class of sub-uniformly distributed sets.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-04-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12910\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12910\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12910","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
更确切地说,粗线条可以覆盖到最小 { t , 3 - t }。 $min \lbrace t,3 - t\rbrace$ 维的部分。我们还考虑了 B $B$ 被 s $s$ 重线覆盖的部分,即那些 dim H ( B ∩ ℓ ) ⩾ s $\dim _{mathrm{H}}(B \cap \ell) \geqslant s$ for s > t - 1 $s &gt; t - 1$ 。我们给出了问题的明确答案:在一般方向上,s $s$ 重线能覆盖多少范围?最后,我们确定了一类新的集合,称为亚均匀分布集合,它们是阿弗斯正则集合的一般化。粗略地说,这些集合与 Ahlfors 不规则集合一样具有空间均匀性,但对不同尺度上的均匀性没有限制。然后,我们将第一作者之前关于阿氏正则集合的结果扩展到次均匀分布集合类,并使之更加清晰。
One formulation of Marstrand's slicing theorem is the following. Assume that , and is a Borel set with . Then, for almost all directions , almost all of is covered by lines parallel to with . We investigate the prospects of sharpening Marstrand's result in the following sense: in a generic direction , is it true that a strictly less than -dimensional part of is covered by the heavy lines , namely those with ? A positive answer for -regular sets was previously obtained by the first author. The answer for general Borel sets turns out to be negative for and positive for . More precisely, the heavy lines can cover up to a dimensional part of in a generic direction. We also consider the part of covered by the -heavy lines, namely those with for . We establish a sharp answer to the question: how much can the -heavy lines cover in a generic direction? Finally, we identify a new class of sets called sub-uniformly distributed sets, which generalise Ahlfors-regular sets. Roughly speaking, these sets share the spatial uniformity of Ahlfors-regular sets, but pose no restrictions on uniformity across different scales. We then extend and sharpen the first author's previous result on Ahlfors-regular sets to the class of sub-uniformly distributed sets.
期刊介绍:
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.