{"title":"接近整数的平方根之和","authors":"Stefan Steinerberger","doi":"10.1016/j.jnt.2024.03.002","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span><math><mi>k</mi><mo>∈</mo><mi>N</mi></math></span> and suppose we are given <em>k</em> integers <span><math><mn>1</mn><mo>≤</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≤</mo><mi>n</mi></math></span>. If <span><math><msqrt><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msqrt><mo>+</mo><mo>…</mo><mo>+</mo><msqrt><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msqrt></math></span> is not an integer, how close can it be to one? When <span><math><mi>k</mi><mo>=</mo><mn>1</mn></math></span>, the distance to the nearest integer is <span><math><mo>≳</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math></span>. Angluin-Eisenstat observed the bound <span><math><mo>≳</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></math></span> when <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span>. We prove there is a universal <span><math><mi>c</mi><mo>></mo><mn>0</mn></math></span> such that, for all <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, there exists a <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>></mo><mn>0</mn></math></span> and <em>k</em> integers in <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span> with<span><span><span><math><mn>0</mn><mo><</mo><mo>‖</mo><msqrt><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msqrt><mo>+</mo><mo>…</mo><mo>+</mo><msqrt><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msqrt><mo>‖</mo><mo>≤</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>⋅</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>c</mi><mo>⋅</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></msup><mo>,</mo></math></span></span></span> where <span><math><mo>‖</mo><mo>⋅</mo><mo>‖</mo></math></span> denotes the distance to the nearest integer. This is a case of the square-root sum problem in numerical analysis where the usual cancellation constructions do not apply: already for <span><math><mi>k</mi><mo>=</mo><mn>3</mn></math></span>, the problem appears hard.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Sums of square roots that are close to an integer\",\"authors\":\"Stefan Steinerberger\",\"doi\":\"10.1016/j.jnt.2024.03.002\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <span><math><mi>k</mi><mo>∈</mo><mi>N</mi></math></span> and suppose we are given <em>k</em> integers <span><math><mn>1</mn><mo>≤</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≤</mo><mi>n</mi></math></span>. If <span><math><msqrt><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msqrt><mo>+</mo><mo>…</mo><mo>+</mo><msqrt><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msqrt></math></span> is not an integer, how close can it be to one? When <span><math><mi>k</mi><mo>=</mo><mn>1</mn></math></span>, the distance to the nearest integer is <span><math><mo>≳</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math></span>. Angluin-Eisenstat observed the bound <span><math><mo>≳</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></math></span> when <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span>. We prove there is a universal <span><math><mi>c</mi><mo>></mo><mn>0</mn></math></span> such that, for all <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, there exists a <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>></mo><mn>0</mn></math></span> and <em>k</em> integers in <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span> with<span><span><span><math><mn>0</mn><mo><</mo><mo>‖</mo><msqrt><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msqrt><mo>+</mo><mo>…</mo><mo>+</mo><msqrt><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msqrt><mo>‖</mo><mo>≤</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>⋅</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>c</mi><mo>⋅</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></msup><mo>,</mo></math></span></span></span> where <span><math><mo>‖</mo><mo>⋅</mo><mo>‖</mo></math></span> denotes the distance to the nearest integer. This is a case of the square-root sum problem in numerical analysis where the usual cancellation constructions do not apply: already for <span><math><mi>k</mi><mo>=</mo><mn>3</mn></math></span>, the problem appears hard.</p></div>\",\"PeriodicalId\":50110,\"journal\":{\"name\":\"Journal of Number Theory\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-04-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Number Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022314X24000763\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Number Theory","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24000763","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Let and suppose we are given k integers . If is not an integer, how close can it be to one? When , the distance to the nearest integer is . Angluin-Eisenstat observed the bound when . We prove there is a universal such that, for all , there exists a and k integers in with where denotes the distance to the nearest integer. This is a case of the square-root sum problem in numerical analysis where the usual cancellation constructions do not apply: already for , the problem appears hard.
期刊介绍:
The Journal of Number Theory (JNT) features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field.
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