{"title":"论厄尔多斯和格雷厄姆关于严格递增序列中连续和的一个问题","authors":"Adrian Beker","doi":"10.1112/blms.13098","DOIUrl":null,"url":null,"abstract":"<p>We show the existence of a constant <span></span><math>\n <semantics>\n <mrow>\n <mi>c</mi>\n <mo>></mo>\n <mn>0</mn>\n </mrow>\n <annotation>$c &gt; 0$</annotation>\n </semantics></math> such that, for all positive integers <span></span><math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math>, there exist integers <span></span><math>\n <semantics>\n <mrow>\n <mn>1</mn>\n <mo>≤</mo>\n <msub>\n <mi>a</mi>\n <mn>1</mn>\n </msub>\n <mo><</mo>\n <mi>⋯</mi>\n <mo><</mo>\n <msub>\n <mi>a</mi>\n <mi>k</mi>\n </msub>\n <mo>≤</mo>\n <mi>n</mi>\n </mrow>\n <annotation>$1 \\leq a_1 &lt; \\cdots &lt; a_k \\leq n$</annotation>\n </semantics></math> such that there are at least <span></span><math>\n <semantics>\n <mrow>\n <mi>c</mi>\n <msup>\n <mi>n</mi>\n <mn>2</mn>\n </msup>\n </mrow>\n <annotation>$cn^2$</annotation>\n </semantics></math> distinct integers of the form <span></span><math>\n <semantics>\n <mrow>\n <msubsup>\n <mo>∑</mo>\n <mrow>\n <mi>i</mi>\n <mo>=</mo>\n <mi>u</mi>\n </mrow>\n <mi>v</mi>\n </msubsup>\n <msub>\n <mi>a</mi>\n <mi>i</mi>\n </msub>\n </mrow>\n <annotation>$\\sum _{i=u}^{v}a_i$</annotation>\n </semantics></math> with <span></span><math>\n <semantics>\n <mrow>\n <mn>1</mn>\n <mo>≤</mo>\n <mi>u</mi>\n <mo>≤</mo>\n <mi>v</mi>\n <mo>≤</mo>\n <mi>k</mi>\n </mrow>\n <annotation>$1 \\leq u \\leq v \\leq k$</annotation>\n </semantics></math>. This answers a question of Erdős and Graham. We also prove a non-trivial upper bound on the maximum number of distinct integers of this form and address several open problems.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 8","pages":"2749-2759"},"PeriodicalIF":0.8000,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On a problem of Erdős and Graham about consecutive sums in strictly increasing sequences\",\"authors\":\"Adrian Beker\",\"doi\":\"10.1112/blms.13098\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We show the existence of a constant <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>c</mi>\\n <mo>></mo>\\n <mn>0</mn>\\n </mrow>\\n <annotation>$c &gt; 0$</annotation>\\n </semantics></math> such that, for all positive integers <span></span><math>\\n <semantics>\\n <mi>n</mi>\\n <annotation>$n$</annotation>\\n </semantics></math>, there exist integers <span></span><math>\\n <semantics>\\n <mrow>\\n <mn>1</mn>\\n <mo>≤</mo>\\n <msub>\\n <mi>a</mi>\\n <mn>1</mn>\\n </msub>\\n <mo><</mo>\\n <mi>⋯</mi>\\n <mo><</mo>\\n <msub>\\n <mi>a</mi>\\n <mi>k</mi>\\n </msub>\\n <mo>≤</mo>\\n <mi>n</mi>\\n </mrow>\\n <annotation>$1 \\\\leq a_1 &lt; \\\\cdots &lt; a_k \\\\leq n$</annotation>\\n </semantics></math> such that there are at least <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>c</mi>\\n <msup>\\n <mi>n</mi>\\n <mn>2</mn>\\n </msup>\\n </mrow>\\n <annotation>$cn^2$</annotation>\\n </semantics></math> distinct integers of the form <span></span><math>\\n <semantics>\\n <mrow>\\n <msubsup>\\n <mo>∑</mo>\\n <mrow>\\n <mi>i</mi>\\n <mo>=</mo>\\n <mi>u</mi>\\n </mrow>\\n <mi>v</mi>\\n </msubsup>\\n <msub>\\n <mi>a</mi>\\n <mi>i</mi>\\n </msub>\\n </mrow>\\n <annotation>$\\\\sum _{i=u}^{v}a_i$</annotation>\\n </semantics></math> with <span></span><math>\\n <semantics>\\n <mrow>\\n <mn>1</mn>\\n <mo>≤</mo>\\n <mi>u</mi>\\n <mo>≤</mo>\\n <mi>v</mi>\\n <mo>≤</mo>\\n <mi>k</mi>\\n </mrow>\\n <annotation>$1 \\\\leq u \\\\leq v \\\\leq k$</annotation>\\n </semantics></math>. This answers a question of Erdős and Graham. We also prove a non-trivial upper bound on the maximum number of distinct integers of this form and address several open problems.</p>\",\"PeriodicalId\":55298,\"journal\":{\"name\":\"Bulletin of the London Mathematical Society\",\"volume\":\"56 8\",\"pages\":\"2749-2759\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-05-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the London Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/blms.13098\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/blms.13098","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我们证明了一个常数 c > 0 $c > 0$ 的存在，即对于所有正整数 n $n$ ，存在整数 1 ≤ a 1 < ⋯ < a k ≤ n $1 \leq a_1 < \cdots < a_k \leq n$ ，这样至少有 c n 2 $cn^2$ 形式为 ∑ i = u v a i $sum _{i=u}^{v}a_i$ 的不同整数，其中 1 ≤ u ≤ v ≤ k $1 \leq u \leq v \leq k$ 。这回答了厄尔多斯和格雷厄姆的一个问题。我们还证明了关于这种形式的不同整数的最大数目的非难上限，并解决了几个悬而未决的问题。
On a problem of Erdős and Graham about consecutive sums in strictly increasing sequences
We show the existence of a constant such that, for all positive integers , there exist integers such that there are at least distinct integers of the form with . This answers a question of Erdős and Graham. We also prove a non-trivial upper bound on the maximum number of distinct integers of this form and address several open problems.
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