{"title":"关于一个近似素数的Piatetski-Shapiro模拟问题","authors":"W.-G. Zhai, Y.-T. Zhao","doi":"10.1007/s10474-023-01371-1","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>N</i> be a sufficiently large number, <span>\\(\\mathfrak{A}\\)</span> and <span>\\(\\mathfrak{B}\\)</span> be subsets of <span>\\(\\{N+1, \\ldots , 2N\\}\\)</span>. We prove that if <span>\\(1<c<\\frac{6}{5}\\)</span>, <span>\\(|\\mathfrak{A}|\\, |\\mathfrak{B}|\\gg N^{2-2\\delta}\\)</span> and <span>\\(\\delta>0\\)</span> is sufficiently small, then the equation\n</p><div><div><span>$$ab=\\lfloor n^c\\rfloor,\\quad a\\in\\mathfrak{A},\\ b\\in\\mathfrak{B}\n$$</span></div></div><p>\n is solvable, which improves the result of Rivat and Sárközy [14]. We also investigate the solvability of the equation\n</p><div><div><span>$$ab=\\lfloor P_k^c\\rfloor,\\quad a\\in\\mathfrak{A},\\ b\\in\\mathfrak{B},\\ 1<c<c_0,\n$$</span></div></div><p>\n \nwhere <i>P</i><sub><i>k</i></sub> denotes an almost-prime with at most <i>k</i> prime factors and <i>c</i><sub>0</sub> is a fixed real number depends on <i>k</i>.\n</p></div>","PeriodicalId":50894,"journal":{"name":"Acta Mathematica Hungarica","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10474-023-01371-1.pdf","citationCount":"0","resultStr":"{\"title\":\"On a Piatetski-Shapiro analog problem over almost-primes\",\"authors\":\"W.-G. Zhai, Y.-T. Zhao\",\"doi\":\"10.1007/s10474-023-01371-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <i>N</i> be a sufficiently large number, <span>\\\\(\\\\mathfrak{A}\\\\)</span> and <span>\\\\(\\\\mathfrak{B}\\\\)</span> be subsets of <span>\\\\(\\\\{N+1, \\\\ldots , 2N\\\\}\\\\)</span>. We prove that if <span>\\\\(1<c<\\\\frac{6}{5}\\\\)</span>, <span>\\\\(|\\\\mathfrak{A}|\\\\, |\\\\mathfrak{B}|\\\\gg N^{2-2\\\\delta}\\\\)</span> and <span>\\\\(\\\\delta>0\\\\)</span> is sufficiently small, then the equation\\n</p><div><div><span>$$ab=\\\\lfloor n^c\\\\rfloor,\\\\quad a\\\\in\\\\mathfrak{A},\\\\ b\\\\in\\\\mathfrak{B}\\n$$</span></div></div><p>\\n is solvable, which improves the result of Rivat and Sárközy [14]. We also investigate the solvability of the equation\\n</p><div><div><span>$$ab=\\\\lfloor P_k^c\\\\rfloor,\\\\quad a\\\\in\\\\mathfrak{A},\\\\ b\\\\in\\\\mathfrak{B},\\\\ 1<c<c_0,\\n$$</span></div></div><p>\\n \\nwhere <i>P</i><sub><i>k</i></sub> denotes an almost-prime with at most <i>k</i> prime factors and <i>c</i><sub>0</sub> is a fixed real number depends on <i>k</i>.\\n</p></div>\",\"PeriodicalId\":50894,\"journal\":{\"name\":\"Acta Mathematica Hungarica\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-09-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s10474-023-01371-1.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematica Hungarica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10474-023-01371-1\",\"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":"Acta Mathematica Hungarica","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10474-023-01371-1","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
On a Piatetski-Shapiro analog problem over almost-primes
Let N be a sufficiently large number, \(\mathfrak{A}\) and \(\mathfrak{B}\) be subsets of \(\{N+1, \ldots , 2N\}\). We prove that if \(1<c<\frac{6}{5}\), \(|\mathfrak{A}|\, |\mathfrak{B}|\gg N^{2-2\delta}\) and \(\delta>0\) is sufficiently small, then the equation
期刊介绍:
Acta Mathematica Hungarica is devoted to publishing research articles of top quality in all areas of pure and applied mathematics as well as in theoretical computer science. The journal is published yearly in three volumes (two issues per volume, in total 6 issues) in both print and electronic formats. Acta Mathematica Hungarica (formerly Acta Mathematica Academiae Scientiarum Hungaricae) was founded in 1950 by the Hungarian Academy of Sciences.