Indagationes Mathematicae-New Series最新文献

{"title":"Cartan–Helgason theorem for quaternionic symmetric and twistor spaces","authors":"Clemens Weiske , Jun Yu , Genkai Zhang","doi":"10.1016/j.indag.2024.05.013","DOIUrl":"10.1016/j.indag.2024.05.013","url":null,"abstract":"<div><div>Let <span><math><mrow><mo>(</mo><mi>g</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span> be a complex quaternionic symmetric pair with <span><math><mi>k</mi></math></span> having an ideal <span><math><mrow><mi>sl</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>k</mi><mo>=</mo><mi>sl</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>ℂ</mi><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math></span>. Consider the representation <span><math><mrow><msup><mrow><mi>S</mi></mrow><mrow><mi>m</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></math></span> of <span><math><mi>k</mi></math></span> via the projection <span><math><mrow><mi>k</mi><mo>→</mo><mi>sl</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span> onto the ideal <span><math><mrow><mi>sl</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span>. We study the finite dimensional irreducible representations <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>g</mi></math></span> which contain <span><math><mrow><msup><mrow><mi>S</mi></mrow><mrow><mi>m</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> under <span><math><mrow><mi>k</mi><mo>⊆</mo><mi>g</mi></mrow></math></span>. We give a characterization of all such representations <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></math></span> and find the corresponding multiplicity, the dimension of <span><math><mrow><mo>Hom</mo><mrow><mo>(</mo><mi>V</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow><msub><mrow><mo>|</mo></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>m</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mo>)</mo></mrow><mo>.</mo></mrow></math></span> We consider also the branching problem of <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></math></span> under <span><math><mrow><mi>l</mi><mo>=</mo><mi>u</mi><msub><mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><mi>ℂ</mi></mrow></msub><mo>+</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>⊂</mo><mi>k</mi></mrow></math></span> and find the multiplicities. Geometrically the Lie subalgebra <span><math><mrow><mi>l</mi><mo>⊂</mo><mi>k</mi></mrow></math></span> defines a twistor space over the compact symmetric space of the compact real form <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>u</mi></mrow></msub></math></span> of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>ℂ</mi></mrow></msub></math></span>, <span><math><mro","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 2","pages":"Pages 357-382"},"PeriodicalIF":0.5,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552929","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":"Submersions, immersions, and étale maps in diffeology","authors":"Alireza Ahmadi","doi":"10.1016/j.indag.2024.03.004","DOIUrl":"10.1016/j.indag.2024.03.004","url":null,"abstract":"<div><p>Although structural maps such as subductions and inductions appear naturally in diffeology, one of the challenges is providing suitable analogues for submersions, immersions, and étale maps (i.e., local diffeomorphisms) consistent with the classical versions of these maps between manifolds. In this paper, we consider diffeological or plotwise versions of submersions, immersions, and étale maps as an adaptation of these maps to diffeology by a nonlinear approach. We study their diffeological properties from different aspects in a systematic fashion with respect to the germs of plots.</p><p>In order to characterize the considered maps from their linear behaviors, we introduce a class of diffeological spaces, so-called diffeological étale manifolds, which not only contains the usual manifolds but also includes irrational tori. We state and prove versions of the rank and implicit function theorems, as well as the fundamental theorem on flows in this class. As an application, we use the results of this work to facilitate the computations of the internal tangent spaces and diffeological dimensions in a few interesting cases.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 3","pages":"Pages 459-499"},"PeriodicalIF":0.6,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140203436","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":"Riesz completions of some spaces of regular operators","authors":"A.W. Wickstead","doi":"10.1016/j.indag.2024.03.002","DOIUrl":"10.1016/j.indag.2024.03.002","url":null,"abstract":"<div><p>We describe the Riesz completion (in the sense of van Haandel) of some spaces of regular operators as explicitly identified subspaces of the regular operators into larger range spaces.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 3","pages":"Pages 443-458"},"PeriodicalIF":0.6,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357724000120/pdfft?md5=fd9d14e2af70c1e89e3e3b2766d40e84&pid=1-s2.0-S0019357724000120-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140150271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On moments and symmetrical sequences","authors":"Jiten Ahuja,&nbsp;Ricardo Estrada","doi":"10.1016/j.indag.2024.04.008","DOIUrl":"10.1016/j.indag.2024.04.008","url":null,"abstract":"&lt;div&gt;&lt;p&gt;In this article we consider questions related to the behavior of the moments &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;z&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;/mfenced&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; when the indices are restricted to specific subsequences of integers, such as the even or odd moments. If &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;n&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 introduce the notion of symmetrical series of order &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; showing that if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;z&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;/mfenced&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is symmetrical then &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;z&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;/mfenced&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; whenever &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;∤&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; in particular, the odd moments of a symmetrical series of order 2 vanish. We prove that when &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;z&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;/mfenced&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; for some &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; then several results characterizing the sequence from its moments hold. We show, in particular, that if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;z&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;/mfenced&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; whenever &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;∤&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; then &lt;span&gt;&lt;math&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;z&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;/mfenced&gt;&lt;/math&gt;&lt;/span&gt; is a rearrangement of a symmetrical series of order &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; We then construct examples of sequences whose moments vanish with required density. Lastly, we construct counterexamples of several of the results valid in the &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; case if we allow the moment series to be all &lt;em&gt;conditionally convergent&lt;/em&gt;. We show that for each &lt;em&gt;arbitrary&lt;/em&gt; sequence of real numbers &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&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;∞&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; there are real sequences &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&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;/mfenced&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&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;∞&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 3","pages":"Pages 584-594"},"PeriodicalIF":0.6,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140928614","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":"Sign involutions on para-abelian varieties","authors":"Jakob Bergqvist,&nbsp;Thuong Dang,&nbsp;Stefan Schröer","doi":"10.1016/j.indag.2024.04.006","DOIUrl":"10.1016/j.indag.2024.04.006","url":null,"abstract":"<div><p>We study the so-called sign involutions on twisted forms of abelian varieties, and show that such a sign involution exists if and only if the class in the Weil–Châtelet group is annihilated by two. If these equivalent conditions hold, we prove that the Picard scheme of the quotient is étale and contains no points of finite order. In dimension one, such quotients are Brauer–Severi curves, and we analyze the ensuing embeddings of the genus-one curve into twisted forms of Hirzebruch surfaces and weighted projective spaces.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 3","pages":"Pages 570-583"},"PeriodicalIF":0.6,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357724000387/pdfft?md5=204d239f9d696a1e77d8dd327376fb09&pid=1-s2.0-S0019357724000387-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140928444","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Explicit dynamical systems on the Sierpiński carpet","authors":"Worapan Homsomboon","doi":"10.1016/j.indag.2024.02.003","DOIUrl":"10.1016/j.indag.2024.02.003","url":null,"abstract":"<div><p>We apply Boroński and Oprocha’s inverse limit construction of dynamical systems on the Sierpiński carpet by using the initial systems of <span><math><mi>n</mi></math></span>-Chamanara surfaces and their <span><math><mi>n</mi></math></span>-baker transformations, <span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></math></span>. We show that all positive real numbers are realized as metric entropy values of dynamical systems on the carpet. We also produce a simplification of Boroński and Oprocha’s proof showing that dynamical systems on the carpet do not have the Bowen specification property.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 3","pages":"Pages 407-433"},"PeriodicalIF":0.6,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139948316","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":"Box dimension of generic Hölder level sets","authors":"Zoltán Buczolich ,&nbsp;Balázs Maga","doi":"10.1016/j.indag.2024.03.015","DOIUrl":"10.1016/j.indag.2024.03.015","url":null,"abstract":"<div><p>Hausdorff dimension of level sets of generic continuous functions defined on fractals can give information about the “thickness/narrow cross-sections” of a “network” corresponding to a fractal set. This leads to the definition of the topological Hausdorff dimension of fractals. Finer information might be obtained by considering the Hausdorff dimension of level sets of generic 1-Hölder-<span><math><mi>α</mi></math></span> functions, which has a stronger dependence on the geometry of the fractal, as displayed in our previous papers (Buczolich et al., 2022 [9,10]). In this paper, we extend our investigations to the lower and upper box-counting dimensions as well: while the former yields results highly resembling the ones about the Hausdorff dimension of level sets, the latter exhibits a different behavior. Instead of “finding narrow-cross sections”, results related to upper box-counting dimension “measure” how much level sets can spread out on the fractal, and how widely the generic function can “oscillate” on it. Key differences are illustrated by giving estimates concerning the Sierpiński triangle.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 3","pages":"Pages 531-554"},"PeriodicalIF":0.6,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357724000326/pdfft?md5=5a77b60c431ef034f802646912c24066&pid=1-s2.0-S0019357724000326-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140589791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"C1-genericity of unbounded distortion for ergodic conservative expanding circle maps","authors":"Hamza Ounesli","doi":"10.1016/j.indag.2024.03.013","DOIUrl":"10.1016/j.indag.2024.03.013","url":null,"abstract":"<div><p>We prove that within the space of ergodic Lebesgue-preserving <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> uniformly expanding maps of the circle, unbounded distortion is <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-generic.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 3","pages":"Pages 523-530"},"PeriodicalIF":0.6,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140796320","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":"Rational points in translations of the Cantor set","authors":"Kan Jiang ,&nbsp;Derong Kong ,&nbsp;Wenxia Li ,&nbsp;Zhiqiang Wang","doi":"10.1016/j.indag.2024.03.012","DOIUrl":"10.1016/j.indag.2024.03.012","url":null,"abstract":"<div><p>Given two coprime integers <span><math><mrow><mi>p</mi><mo>≥</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>q</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, let <span><math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>⊂</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> consist of all rational numbers which have a finite <span><math><mi>p</mi></math></span>-ary expansion, and let <span><span><span><math><mrow><mi>K</mi><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow><mo>=</mo><mfenced><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></munderover><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow></msup></mrow></mfrac><mo>:</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>A</mi><mspace></mspace><mo>∀</mo><mi>i</mi><mo>∈</mo><mi>N</mi></mrow></mfenced><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mi>A</mi><mo>⊂</mo><mfenced><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>q</mi><mo>−</mo><mn>1</mn></mrow></mfenced></mrow></math></span> with cardinality <span><math><mrow><mn>1</mn><mo>&lt;</mo><mi>#</mi><mi>A</mi><mo>&lt;</mo><mi>q</mi></mrow></math></span>. In 2021 Schleischitz showed that <span><math><mrow><mi>#</mi><mrow><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>∩</mo><mi>K</mi><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>&lt;</mo><mo>+</mo><mi>∞</mi></mrow></math></span>. In this paper we show that for any <span><math><mrow><mi>r</mi><mo>∈</mo><mi>Q</mi></mrow></math></span> and for any <span><math><mrow><mi>α</mi><mo>∈</mo><mi>R</mi></mrow></math></span>, <span><span><span><math><mrow><mi>#</mi><mrow><mo>(</mo><mrow><mrow><mo>(</mo><mi>r</mi><msub><mrow><mi>D</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>+</mo><mi>α</mi><mo>)</mo></mrow><mo>∩</mo><mi>K</mi><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>&lt;</mo><mo>+</mo><mi>∞</mi><mo>.</mo></mrow></math></span></span></span></p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 3","pages":"Pages 516-522"},"PeriodicalIF":0.6,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799303","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 perfect powers that are sums of cubes of a nine term arithmetic progression","authors":"Nirvana Coppola ,&nbsp;Mar Curcó-Iranzo ,&nbsp;Maleeha Khawaja ,&nbsp;Vandita Patel ,&nbsp;Özge Ülkem","doi":"10.1016/j.indag.2024.03.011","DOIUrl":"10.1016/j.indag.2024.03.011","url":null,"abstract":"<div><p>We study the equation <span><math><mrow><msup><mrow><mrow><mo>(</mo><mi>x</mi><mo>−</mo><mn>4</mn><mi>r</mi><mo>)</mo></mrow></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mi>x</mi><mo>−</mo><mn>3</mn><mi>r</mi><mo>)</mo></mrow></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mi>r</mi><mo>)</mo></mrow></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mi>x</mi><mo>−</mo><mi>r</mi><mo>)</mo></mrow></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>r</mi><mo>)</mo></mrow></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mi>r</mi><mo>)</mo></mrow></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>3</mn><mi>r</mi><mo>)</mo></mrow></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>4</mn><mi>r</mi><mo>)</mo></mrow></mrow><mrow><mn>3</mn></mrow></msup><mo>=</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>p</mi></mrow></msup></mrow></math></span>, which is a natural continuation of previous works carried out by A. Argáez-García and the fourth author (perfect powers that are sums of cubes of a three, five and seven term arithmetic progression). Under the assumptions <span><math><mrow><mn>0</mn><mo>&lt;</mo><mi>r</mi><mo>≤</mo><mn>1</mn><msup><mrow><mn>0</mn></mrow><mrow><mn>6</mn></mrow></msup></mrow></math></span>, <span><math><mrow><mi>p</mi><mo>≥</mo><mn>5</mn></mrow></math></span> a prime and <span><math><mrow><mo>gcd</mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span>, we show that solutions must satisfy <span><math><mrow><mi>x</mi><mi>y</mi><mo>=</mo><mn>0</mn></mrow></math></span>. Moreover, we study the equation for prime exponents 2 and 3 in greater detail. Under the assumptions <span><math><mrow><mi>r</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span> a positive integer and <span><math><mrow><mo>gcd</mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> we show that there are infinitely many solutions for <span><math><mrow><mi>p</mi><mo>=</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>p</mi><mo>=</mo><mn>3</mn></mrow></math></span> via explicit constructions using integral points on elliptic curves. We use an amalgamation of methods in computational and algebraic number theory to overcome the increased computational challenge. Most notable is a significant computational efficiency obtained through appealing to Bilu, Hanrot and Voutier’s Primitive Divisor Theorem and the method of Chabauty, as well as employing a Thue equation solver earlier on.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 3","pages":"Pages 500-515"},"PeriodicalIF":0.6,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357724000296/pdfft?md5=883869d8b3a6f3a8bbf7ff0b2b89d307&pid=1-s2.0-S0019357724000296-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140806441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}