Dimitrios Charamaras, Andreas Mountakis, Konstantinos Tsinas
{"title":"关于沿线性模式的乘法递归","authors":"Dimitrios Charamaras, Andreas Mountakis, Konstantinos Tsinas","doi":"10.1112/jlms.70292","DOIUrl":null,"url":null,"abstract":"<p>Donoso, Le, Moreira, and Sun (<i>J. Anal. Math</i>. 149 (2023), 719–761) study sets of recurrence for actions of the multiplicative semigroup <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>N</mi>\n <mo>,</mo>\n <mo>×</mo>\n <mo>)</mo>\n </mrow>\n <annotation>$(\\mathbb {N}, \\times)$</annotation>\n </semantics></math> and provide some sufficient conditions for sets of the form <span></span><math>\n <semantics>\n <mrow>\n <mi>S</mi>\n <mo>=</mo>\n <mo>{</mo>\n <mo>(</mo>\n <mi>a</mi>\n <mi>n</mi>\n <mo>+</mo>\n <mi>b</mi>\n <mo>)</mo>\n <mo>/</mo>\n <mo>(</mo>\n <mi>c</mi>\n <mi>n</mi>\n <mo>+</mo>\n <mi>d</mi>\n <mo>)</mo>\n <mo>:</mo>\n <mi>n</mi>\n <mo>∈</mo>\n <mi>N</mi>\n <mo>}</mo>\n </mrow>\n <annotation>$S=\\lbrace (an+b)/(cn+d) \\colon n \\in \\mathbb {N}\\rbrace $</annotation>\n </semantics></math> to be sets of recurrence for such actions. A necessary condition for <span></span><math>\n <semantics>\n <mi>S</mi>\n <annotation>$S$</annotation>\n </semantics></math> to be a set of multiplicative recurrence is that for every completely multiplicative function <span></span><math>\n <semantics>\n <mi>f</mi>\n <annotation>$f$</annotation>\n </semantics></math> taking values on the unit circle, we have that <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mo>lim inf</mo>\n <mrow>\n <mi>n</mi>\n <mo>→</mo>\n <mi>∞</mi>\n </mrow>\n </msub>\n <mrow>\n <mo>|</mo>\n <mi>f</mi>\n <mrow>\n <mo>(</mo>\n <mi>a</mi>\n <mi>n</mi>\n <mo>+</mo>\n <mi>b</mi>\n <mo>)</mo>\n </mrow>\n <mo>−</mo>\n <mi>f</mi>\n <mrow>\n <mo>(</mo>\n <mi>c</mi>\n <mi>n</mi>\n <mo>+</mo>\n <mi>d</mi>\n <mo>)</mo>\n </mrow>\n <mo>|</mo>\n </mrow>\n <mo>=</mo>\n <mn>0</mn>\n </mrow>\n <annotation>$\\liminf _{n \\rightarrow \\infty } |f(an+b)-f(cn+d)|=0$</annotation>\n </semantics></math>. In this article, we fully characterize the integer quadruples <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>a</mi>\n <mo>,</mo>\n <mi>b</mi>\n <mo>,</mo>\n <mi>c</mi>\n <mo>,</mo>\n <mi>d</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$(a,b,c,d)$</annotation>\n </semantics></math> which satisfy the latter property. Our result generalizes a result of Klurman and Mangerel (<i>Math. Ann</i>. 372 (2018), 651–697) concerning the pair <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>n</mi>\n <mo>,</mo>\n <mi>n</mi>\n <mo>+</mo>\n <mn>1</mn>\n <mo>)</mo>\n </mrow>\n <annotation>$(n,n+1)$</annotation>\n </semantics></math>, as well as some results from (Donoso, Le, Moreira, and Sun, <i>J. Anal. Math</i>. 149 (2023), 719–761). In addition, we prove that, under the same conditions on <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>a</mi>\n <mo>,</mo>\n <mi>b</mi>\n <mo>,</mo>\n <mi>c</mi>\n <mo>,</mo>\n <mi>d</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$(a,b,c,d)$</annotation>\n </semantics></math>, the set <span></span><math>\n <semantics>\n <mi>S</mi>\n <annotation>$S$</annotation>\n </semantics></math> is a set of recurrence for finitely generated actions of <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>N</mi>\n <mo>,</mo>\n <mo>×</mo>\n <mo>)</mo>\n </mrow>\n <annotation>$(\\mathbb {N}, \\times)$</annotation>\n </semantics></math>.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"112 3","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2025-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70292","citationCount":"0","resultStr":"{\"title\":\"On multiplicative recurrence along linear patterns\",\"authors\":\"Dimitrios Charamaras, Andreas Mountakis, Konstantinos Tsinas\",\"doi\":\"10.1112/jlms.70292\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Donoso, Le, Moreira, and Sun (<i>J. Anal. Math</i>. 149 (2023), 719–761) study sets of recurrence for actions of the multiplicative semigroup <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mi>N</mi>\\n <mo>,</mo>\\n <mo>×</mo>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(\\\\mathbb {N}, \\\\times)$</annotation>\\n </semantics></math> and provide some sufficient conditions for sets of the form <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>S</mi>\\n <mo>=</mo>\\n <mo>{</mo>\\n <mo>(</mo>\\n <mi>a</mi>\\n <mi>n</mi>\\n <mo>+</mo>\\n <mi>b</mi>\\n <mo>)</mo>\\n <mo>/</mo>\\n <mo>(</mo>\\n <mi>c</mi>\\n <mi>n</mi>\\n <mo>+</mo>\\n <mi>d</mi>\\n <mo>)</mo>\\n <mo>:</mo>\\n <mi>n</mi>\\n <mo>∈</mo>\\n <mi>N</mi>\\n <mo>}</mo>\\n </mrow>\\n <annotation>$S=\\\\lbrace (an+b)/(cn+d) \\\\colon n \\\\in \\\\mathbb {N}\\\\rbrace $</annotation>\\n </semantics></math> to be sets of recurrence for such actions. A necessary condition for <span></span><math>\\n <semantics>\\n <mi>S</mi>\\n <annotation>$S$</annotation>\\n </semantics></math> to be a set of multiplicative recurrence is that for every completely multiplicative function <span></span><math>\\n <semantics>\\n <mi>f</mi>\\n <annotation>$f$</annotation>\\n </semantics></math> taking values on the unit circle, we have that <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mo>lim inf</mo>\\n <mrow>\\n <mi>n</mi>\\n <mo>→</mo>\\n <mi>∞</mi>\\n </mrow>\\n </msub>\\n <mrow>\\n <mo>|</mo>\\n <mi>f</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>a</mi>\\n <mi>n</mi>\\n <mo>+</mo>\\n <mi>b</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>−</mo>\\n <mi>f</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>c</mi>\\n <mi>n</mi>\\n <mo>+</mo>\\n <mi>d</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>|</mo>\\n </mrow>\\n <mo>=</mo>\\n <mn>0</mn>\\n </mrow>\\n <annotation>$\\\\liminf _{n \\\\rightarrow \\\\infty } |f(an+b)-f(cn+d)|=0$</annotation>\\n </semantics></math>. In this article, we fully characterize the integer quadruples <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mi>a</mi>\\n <mo>,</mo>\\n <mi>b</mi>\\n <mo>,</mo>\\n <mi>c</mi>\\n <mo>,</mo>\\n <mi>d</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(a,b,c,d)$</annotation>\\n </semantics></math> which satisfy the latter property. Our result generalizes a result of Klurman and Mangerel (<i>Math. Ann</i>. 372 (2018), 651–697) concerning the pair <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mi>n</mi>\\n <mo>,</mo>\\n <mi>n</mi>\\n <mo>+</mo>\\n <mn>1</mn>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(n,n+1)$</annotation>\\n </semantics></math>, as well as some results from (Donoso, Le, Moreira, and Sun, <i>J. Anal. Math</i>. 149 (2023), 719–761). In addition, we prove that, under the same conditions on <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mi>a</mi>\\n <mo>,</mo>\\n <mi>b</mi>\\n <mo>,</mo>\\n <mi>c</mi>\\n <mo>,</mo>\\n <mi>d</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(a,b,c,d)$</annotation>\\n </semantics></math>, the set <span></span><math>\\n <semantics>\\n <mi>S</mi>\\n <annotation>$S$</annotation>\\n </semantics></math> is a set of recurrence for finitely generated actions of <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mi>N</mi>\\n <mo>,</mo>\\n <mo>×</mo>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(\\\\mathbb {N}, \\\\times)$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":\"112 3\",\"pages\":\"\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2025-09-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70292\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70292\",\"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://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70292","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
Donoso, Le， Moreira和Sun （J. Anal）。数学。149(2023),719-761)研究乘法半群(N；×) $(\mathbb {N}, \times)$，并给出S = {(an + b) / (c n)形式的集合的一些充分条件+ d): n∈n}$S=\lbrace (an+b)/(cn+d) \colon n \in \mathbb {N}\rbrace $为这些动作的递归集。S $S$是一组乘法递归式的必要条件是对于每一个在单位圆上取值的完全乘法函数f $f$，我们有lim n→∞| f (a n + b)−f (c n + d) | = 0 $\liminf _{n \rightarrow \infty } |f(an+b)-f(cn+d)|=0$。在本文中，我们完全刻画了满足后一个性质的整数四元组（a, b, c, d） $(a,b,c,d)$。我们的结果推广了Klurman和Mangerel（数学）的结果。Ann. 372(2018), 651-697)关于对（n, n + 1） $(n,n+1)$，以及（Donoso, Le, Moreira, and Sun， J. Anal.）的一些结果。数学。149(2023),719-761)。此外，证明了在（a, b, c, d） $(a,b,c,d)$相同条件下，集合S $S$是有限生成动作（N, x） $(\mathbb {N}, \times)$的递归集。
On multiplicative recurrence along linear patterns
Donoso, Le, Moreira, and Sun (J. Anal. Math. 149 (2023), 719–761) study sets of recurrence for actions of the multiplicative semigroup and provide some sufficient conditions for sets of the form to be sets of recurrence for such actions. A necessary condition for to be a set of multiplicative recurrence is that for every completely multiplicative function taking values on the unit circle, we have that . In this article, we fully characterize the integer quadruples which satisfy the latter property. Our result generalizes a result of Klurman and Mangerel (Math. Ann. 372 (2018), 651–697) concerning the pair , as well as some results from (Donoso, Le, Moreira, and Sun, J. Anal. Math. 149 (2023), 719–761). In addition, we prove that, under the same conditions on , the set is a set of recurrence for finitely generated actions of .
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