Indagationes Mathematicae-New Series最新文献

{"title":"","authors":"Wolter Groenevelt (Guest editors), Erik Koelink, Hjalmar Rosengren, Jasper Stokman","doi":"10.1016/j.indag.2025.09.003","DOIUrl":"10.1016/j.indag.2025.09.003","url":null,"abstract":"","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 6","pages":"Pages 1491-1492"},"PeriodicalIF":0.8,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145374404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Conditions for the difference set of a central Cantor set to be a Cantorval. Part II","authors":"Piotr Nowakowski","doi":"10.1016/j.indag.2025.03.005","DOIUrl":"10.1016/j.indag.2025.03.005","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow><mo>⊂</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></math></span> be the central Cantor set generated by a sequence <span><math><mrow><mi>a</mi><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mfenced><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mi>N</mi></mrow></msup></mrow></math></span>. It is known that the difference set <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow><mo>−</mo><mi>C</mi><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow></math></span> has one of three possible forms: a finite union of closed intervals, a Cantor set, or a Cantorval. In the previous paper (Filipczak and Nowakowski, 2023), there was given the sufficient condition for the sequence <span><math><mi>a</mi></math></span>, which implies that <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow><mo>−</mo><mi>C</mi><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow></math></span> is a Cantorval. In this paper we give different conditions for a sequence <span><math><mi>a</mi></math></span>, which guarantee the same assertion. We also prove a corollary, which provides infinitely many new examples of Cantorvals.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 5","pages":"Pages 1223-1244"},"PeriodicalIF":0.8,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144896430","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On moments of the error term of the multivariable kth divisor functions","authors":"Zhen Guo,&nbsp;Xin Li","doi":"10.1016/j.indag.2025.01.003","DOIUrl":"10.1016/j.indag.2025.01.003","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Suppose &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;⩾&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is an integer. Let &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;τ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; be the number of ways &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; can be written as a product of &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; fixed factors. For any fixed integer &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;⩾&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, we have the asymptotic formula &lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;munder&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⩽&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;τ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;munderover&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;log&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;ɛ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; are computable constants. In this paper we study the mean square of &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and give upper bounds for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;⩾&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and an asymptotic formula for the mean square of &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. We also get an upper bound for the third power moment of &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Moreover, we study the first power moment of &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and then give a result for the sign ","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 5","pages":"Pages 1133-1156"},"PeriodicalIF":0.8,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144896428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Pervasiveness of Lr(E,F) in Lr(E,Fδ)","authors":"Quinn Kiervin Starkey,&nbsp;Foivos Xanthos","doi":"10.1016/j.indag.2025.05.003","DOIUrl":"10.1016/j.indag.2025.05.003","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>E</mi><mo>,</mo><mi>F</mi></mrow></math></span> be Archimedean Riesz spaces, and let <span><math><msup><mrow><mi>F</mi></mrow><mrow><mi>δ</mi></mrow></msup></math></span> denote an order completion of <span><math><mi>F</mi></math></span>. In this note, we provide necessary conditions under which the space of all regular operators <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>r</mi></mrow></msup><mrow><mo>(</mo><mi>E</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> is pervasive in <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>r</mi></mrow></msup><mrow><mo>(</mo><mi>E</mi><mo>,</mo><msup><mrow><mi>F</mi></mrow><mrow><mi>δ</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>. Pervasiveness of <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>r</mi></mrow></msup><mrow><mo>(</mo><mi>E</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> in <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>r</mi></mrow></msup><mrow><mo>(</mo><mi>E</mi><mo>,</mo><msup><mrow><mi>F</mi></mrow><mrow><mi>δ</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> implies that the Riesz completion of <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>r</mi></mrow></msup><mrow><mo>(</mo><mi>E</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> can be realized as a Riesz subspace of <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>r</mi></mrow></msup><mrow><mo>(</mo><mi>E</mi><mo>,</mo><msup><mrow><mi>F</mi></mrow><mrow><mi>δ</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>. It also ensures that the regular part of the space of order continuous operators <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>o</mi><mi>c</mi></mrow></msup><mrow><mo>(</mo><mi>E</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> forms a band of <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>r</mi></mrow></msup><mrow><mo>(</mo><mi>E</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span>. Furthermore, the positive part <span><math><msup><mrow><mi>T</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> of any operator <span><math><mrow><mi>T</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>r</mi></mrow></msup><mrow><mo>(</mo><mi>E</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span>, provided it exists, is given by the Riesz–Kantorovich formula. The results apply in particular to cases where <span><math><mrow><mi>E</mi><mo>=</mo><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mn>0</mn></mrow><mrow><mi>∞</mi></mrow></msubsup></mrow></math></span>, <span><math><mrow><mi>E</mi><mo>=</mo><mi>c</mi></mrow></math></span>, or <span><math><mi>F</mi></math></span> is atomic, and they provide solutions to some problems posed in Abramovich and Wickstead (1991) and Wickstead (2024).</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 5","pages":"Pages 1405-1416"},"PeriodicalIF":0.8,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144895929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A note on the distribution of clusters and deserts of prime numbers","authors":"Eugenio P. Balanzario","doi":"10.1016/j.indag.2025.03.007","DOIUrl":"10.1016/j.indag.2025.03.007","url":null,"abstract":"<div><div>We consider the distribution of values of weighted sums of the von Mangoldt arithmetical function. Using a formula for the distribution of values of trigonometric polynomials, we are able to present evidence supporting the claim that these weighted sums follow a distribution with a normal-like behavior.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 5","pages":"Pages 1276-1287"},"PeriodicalIF":0.8,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144896431","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A graph-theoretic proof of Cobham’s Dichotomy for automatic sequences","authors":"Mieke Wessel","doi":"10.1016/j.indag.2025.04.001","DOIUrl":"10.1016/j.indag.2025.04.001","url":null,"abstract":"<div><div>We give a new graph-theoretic proof of a theorem of Cobham which says that the support of an automatic sequence is either sparse, that is, grows polylogarithmically, or grows at least like <span><math><msup><mrow><mi>N</mi></mrow><mrow><mi>α</mi></mrow></msup></math></span> for some <span><math><mrow><mi>α</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>. The proof uses the notions of tied vertices and cycle arborescences. With the ideas of the proof we can also give a new interpretation of the rank of a sparse sequence as the height of its cycle arborescence. In the non-sparse case we are able to show that the support has asymptotic behavior of the form <span><math><mrow><msup><mrow><mi>N</mi></mrow><mrow><mi>B</mi></mrow></msup><mo>log</mo><msup><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span>, where <span><math><mi>B</mi></math></span> turns out to be the logarithm of an integer root of a Perron number.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 5","pages":"Pages 1310-1328"},"PeriodicalIF":0.8,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144896433","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An elementary proof of the Benjamini–Nekrashevych–Pete conjecture for the semi-direct products Zn⋊Z","authors":"Dean Wardell","doi":"10.1016/j.indag.2025.01.002","DOIUrl":"10.1016/j.indag.2025.01.002","url":null,"abstract":"<div><div>A finitely generated group <span><math><mi>G</mi></math></span> is called strongly scale-invariant if there exists an injective homomorphism <span><math><mrow><mi>f</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>G</mi></mrow></math></span> such that <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is a finite index subgroup of <span><math><mi>G</mi></math></span> and such that <span><math><mrow><msub><mrow><mo>∩</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub><msup><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is finite. Nekrashevych and Pete conjectured that all strongly scale-invariant groups are virtually nilpotent, after disproving a stronger conjecture by Benjamini.</div><div>This conjecture is known to be true in some situations. Deré proved it for virtually polycyclic groups. In this paper, we provide an elementary proof for those polycyclic groups that can be written as a semi-direct product <span><math><mrow><msup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>⋊</mo><mi>Z</mi></mrow></math></span>.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 5","pages":"Pages 1125-1132"},"PeriodicalIF":0.8,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144896425","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Square functions associated with RittE operators","authors":"Oualid Bouabdillah","doi":"10.1016/j.indag.2025.05.014","DOIUrl":"10.1016/j.indag.2025.05.014","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In a finite subset &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ξ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ξ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; of the torus &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, the notion of Ritt&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; operators on a Banach space and their functional calculus on generalized Stolz domains was developed and studied in Bouabdillah and Le Merdy (2024).&lt;/div&gt;&lt;div&gt;In this paper, we define a quadratic functional calculus for a Ritt&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; operator on &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, by a decomposition of type Franks–McIntosh. We show that with some hypothesis on the cotype of &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, this notion is equivalent to the existence of a bounded functional calculus on &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;We define for a Ritt&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; operator on a Banach space &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and for any positive real number &lt;span&gt;&lt;math&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and for any &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;\u0000 &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munder&gt;&lt;mrow&gt;&lt;mo&gt;lim&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt;&lt;munderover&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ɛ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⊗&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;munderover&gt;&lt;mrow&gt;&lt;mo&gt;∏&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ξ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mo&gt;¯&lt;/mo&gt;&lt;/mover&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Rad&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; We show that, under the condition of finite cotype of &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, a Ritt&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; operator admits a quadratic functional calculus if and only if the estimates &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 5","pages":"Pages 1417-1452"},"PeriodicalIF":0.8,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144895930","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Non-self-intersective Dragon curves","authors":"Shigeki Akiyama ,&nbsp;Yuichi Kamiya ,&nbsp;Fan Wen","doi":"10.1016/j.indag.2025.03.006","DOIUrl":"10.1016/j.indag.2025.03.006","url":null,"abstract":"<div><div>Let us fold a strip of paper many times in the same direction, and then unfold it to form a fixed angle <span><math><mi>θ</mi></math></span> at all creases. The resulting shape is called the Dragon curve with the unfolding angle <span><math><mi>θ</mi></math></span>. When <span><math><mrow><mn>0</mn><mo>≤</mo><mi>θ</mi><mo>&lt;</mo><mn>90</mn><mo>°</mo></mrow></math></span>, the corresponding Dragon curve has a self-intersection. When <span><math><mrow><mi>θ</mi><mo>=</mo><mn>180</mn><mo>°</mo></mrow></math></span>, the corresponding Dragon curve is a straight line, which has no self-intersection. In this paper, we will show that any Dragon curve whose unfolding angle is greater than <span><math><mrow><mn>99</mn><mo>.</mo><mn>3438</mn><mo>°</mo></mrow></math></span> and less than <span><math><mrow><mn>180</mn><mo>°</mo></mrow></math></span> has no self-intersection.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 5","pages":"Pages 1245-1275"},"PeriodicalIF":0.8,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144895927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Binary quadratic forms with the same value set","authors":"Étienne Fouvry ,&nbsp;Peter Koymans","doi":"10.1016/j.indag.2025.01.005","DOIUrl":"10.1016/j.indag.2025.01.005","url":null,"abstract":"<div><div>Given a binary quadratic form <span><math><mrow><mi>F</mi><mo>∈</mo><mi>Z</mi><mrow><mo>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>]</mo></mrow></mrow></math></span>, we define its value set <span><math><mrow><mi>F</mi><mrow><mo>(</mo><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> to be <span><math><mrow><mo>{</mo><mi>F</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>:</mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>}</mo></mrow></math></span>. If <span><math><mrow><mi>F</mi><mo>,</mo><mi>G</mi><mo>∈</mo><mi>Z</mi><mrow><mo>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>]</mo></mrow></mrow></math></span> are two binary quadratic forms, we give necessary and sufficient conditions on <span><math><mi>F</mi></math></span> and <span><math><mi>G</mi></math></span> for <span><math><mrow><mi>F</mi><mrow><mo>(</mo><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mo>=</mo><mi>G</mi><mrow><mo>(</mo><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 5","pages":"Pages 1157-1179"},"PeriodicalIF":0.8,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144896426","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}