Indagationes Mathematicae-New Series最新文献

{"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":null,"pages":null},"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":"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":null,"pages":null},"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}

{"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":null,"pages":null},"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":"Root numbers of a family of elliptic curves and two applications","authors":"Jonathan Love","doi":"10.1016/j.indag.2024.04.003","DOIUrl":"10.1016/j.indag.2024.04.003","url":null,"abstract":"<div><p>For each <span><math><mrow><mi>t</mi><mo>∈</mo><mi>Q</mi><mo>∖</mo><mrow><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span>, define an elliptic curve over <span><math><mi>Q</mi></math></span> by <span><span><span><math><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>:</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi>x</mi><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>+</mo><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mo>.</mo></mrow></math></span></span></span>Using a formula for the root number <span><math><mrow><mi>W</mi><mrow><mo>(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> as a function of <span><math><mi>t</mi></math></span> and assuming some standard conjectures about ranks of elliptic curves, we determine (up to a set of density zero) the set of isomorphism classes of elliptic curves <span><math><mrow><mi>E</mi><mo>/</mo><mi>Q</mi></mrow></math></span> whose Mordell–Weil group contains <span><math><mrow><mi>Z</mi><mo>×</mo><mi>Z</mi><mo>/</mo><mn>2</mn><mi>Z</mi><mo>×</mo><mi>Z</mi><mo>/</mo><mn>4</mn><mi>Z</mi></mrow></math></span>, and the set of rational numbers that can be written as a product of the slopes of two rational right triangles.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357724000351/pdfft?md5=2bd90ba3afb1d531934bbb073c1710e2&pid=1-s2.0-S0019357724000351-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799683","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":"Each friend of 10 has at least 10 nonidentical prime factors","authors":"Henry (Maya) Robert Thackeray","doi":"10.1016/j.indag.2024.04.011","DOIUrl":"10.1016/j.indag.2024.04.011","url":null,"abstract":"<div><p>For each positive integer <span><math><mi>n</mi></math></span>, if the sum of the factors of <span><math><mi>n</mi></math></span> is divided by <span><math><mi>n</mi></math></span>, then the result is called the abundancy index of <span><math><mi>n</mi></math></span>. If the abundancy index of some positive integer <span><math><mi>m</mi></math></span> equals the abundancy index of <span><math><mi>n</mi></math></span> but <span><math><mi>m</mi></math></span> is not equal to <span><math><mi>n</mi></math></span>, then <span><math><mi>m</mi></math></span> and <span><math><mi>n</mi></math></span> are called friends. A positive integer with no friends is called solitary. The smallest positive integer that is not known to have a friend and is not known to be solitary is 10.</p><p>It is not known if the number 6 has odd friends, that is, if odd perfect numbers exist. In a 2007 article, Nielsen proved that the number of nonidentical prime factors in any odd perfect number is at least 9. A 2015 article by Nielsen, which was more complicated and used a computer program that took months to complete, increased the lower bound from 9 to 10.</p><p>This work applies methods from Nielsen’s 2007 article to show that each friend of 10 has at least 10 nonidentical prime factors.</p><p>This is a formal write-up of results presented at the Southern Africa Mathematical Sciences Association Conference 2023 at the University of Pretoria.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357724000430/pdfft?md5=5f3ad739533e1db88fb550301881c997&pid=1-s2.0-S0019357724000430-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141060205","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 the separation of the roots of the generalized Fibonacci polynomial","authors":"Jonathan García ,&nbsp;Carlos A. Gómez ,&nbsp;Florian Luca","doi":"10.1016/j.indag.2023.12.002","DOIUrl":"10.1016/j.indag.2023.12.002","url":null,"abstract":"<div><p>In this paper we prove some separation results for the roots of the generalized Fibonacci polynomials and their absolute values.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138679718","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 Diophantine equations 2ln2=1+q+⋯+qα and application to odd perfect numbers","authors":"Yoshinosuke Hirakawa","doi":"10.1016/j.indag.2023.12.004","DOIUrl":"10.1016/j.indag.2023.12.004","url":null,"abstract":"<div><p>Let <span><math><mi>N</mi></math></span> be an odd perfect number. Then, Euler proved that there exist some integers <span><math><mrow><mi>n</mi><mo>,</mo><mi>α</mi></mrow></math></span> and a prime <span><math><mi>q</mi></math></span> such that <span><math><mrow><mi>N</mi><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>q</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></math></span>, <span><math><mrow><mi>q</mi><mo>∤</mo><mi>n</mi></mrow></math></span>, and <span><math><mrow><mi>q</mi><mo>≡</mo><mi>α</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mo>mod</mo><mspace></mspace><mn>4</mn></mrow></math></span>. In this note, we prove that the ratio <span><math><mfrac><mrow><mi>σ</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac></math></span> is neither a square nor a square times a single prime unless <span><math><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow></math></span><span>. It is a direct consequence of a certain property of the Diophantine equation </span><span><math><mrow><mn>2</mn><mi>l</mi><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>1</mn><mo>+</mo><mi>q</mi><mo>+</mo><mo>⋯</mo><mo>+</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></math></span>, where <span><math><mi>l</mi></math></span><span> denotes one or a prime, and its proof is based on the prime ideal factorization in the quadratic orders </span><span><math><mrow><mi>Z</mi><mrow><mo>[</mo><msqrt><mrow><mn>1</mn><mo>−</mo><mi>q</mi></mrow></msqrt><mo>]</mo></mrow></mrow></math></span> and the primitive solutions of generalized Fermat equations <span><math><mrow><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup><mo>+</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>β</mi></mrow></msup><mo>=</mo><mn>2</mn><msup><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span>. We give also a slight generalization to odd multiply perfect numbers.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139104891","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 some coefficients of the Artin–Hasse series modulo a prime","authors":"Marina Avitabile,&nbsp;Sandro Mattarei","doi":"10.1016/j.indag.2024.01.003","DOIUrl":"10.1016/j.indag.2024.01.003","url":null,"abstract":"<div><p>Let <span><math><mi>p</mi></math></span> be an odd prime, and let <span><math><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>∞</mi></mrow></msubsup><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>[</mo><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow><mo>]</mo></mrow></mrow></math></span> be the reduction modulo <span><math><mi>p</mi></math></span> of the Artin–Hasse exponential series. We obtain a polynomial expression for <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi><mi>p</mi></mrow></msub></math></span> in terms of those <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>r</mi><mi>p</mi></mrow></msub></math></span> with <span><math><mrow><mi>r</mi><mo>&lt;</mo><mi>k</mi></mrow></math></span>, for even <span><math><mrow><mi>k</mi><mo>&lt;</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow></math></span>. A conjectural analogue covering the case of odd <span><math><mrow><mi>k</mi><mo>&lt;</mo><mi>p</mi></mrow></math></span> can be stated in various polynomial forms, essentially in terms of the polynomial <span><math><mrow><mi>γ</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>/</mo><mi>n</mi><mo>)</mo></mrow><msup><mrow><mi>X</mi></mrow><mrow><mi>p</mi><mo>−</mo><mi>n</mi></mrow></msup></mrow></math></span>, where <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> denotes the <span><math><mi>n</mi></math></span>th Bernoulli number.</p><p>We prove that <span><math><mrow><mi>γ</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> satisfies the functional equation <span><math><mrow><mi>γ</mi><mrow><mo>(</mo><mi>X</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>−</mo><mi>γ</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>£</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>+</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>−</mo><mn>1</mn></mrow></math></span> in <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow></mrow></math></span>, where <span><math><mrow><msub><mrow><mi>£</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> are the truncated logarithm and the Wilson quotient. This is an analogue modulo <span><math><mi>p</mi></math></span> of a functional equation, in <span>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139506915","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 threshold for the best two-term underapproximation by Egyptian fractions","authors":"Hùng Việt Chu","doi":"10.1016/j.indag.2024.01.006","DOIUrl":"10.1016/j.indag.2024.01.006","url":null,"abstract":"<div><p>Let <span><math><mi>G</mi></math></span><span> be the greedy algorithm that, for each </span><span><math><mrow><mi>θ</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></math></span>, produces an infinite sequence of positive integers <span><math><msubsup><mrow><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></msubsup></math></span> satisfying <span><math><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></msubsup><mn>1</mn><mo>/</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>θ</mi></mrow></math></span>. For natural numbers <span><math><mrow><mi>p</mi><mo>&lt;</mo><mi>q</mi></mrow></math></span>, let <span><math><mrow><mi>Υ</mi><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span><span> denote the smallest positive integer </span><span><math><mi>j</mi></math></span> such that <span><math><mi>p</mi></math></span> divides <span><math><mrow><mi>q</mi><mo>+</mo><mi>j</mi></mrow></math></span>. Continuing Nathanson’s study of two-term underapproximations, we show that whenever <span><math><mrow><mi>Υ</mi><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow><mo>⩽</mo><mn>3</mn></mrow></math></span>, <span><math><mi>G</mi></math></span> gives the (unique) best two-term underapproximation of <span><math><mrow><mi>p</mi><mo>/</mo><mi>q</mi></mrow></math></span>; i.e., if <span><math><mrow><mn>1</mn><mo>/</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>1</mn><mo>/</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>&lt;</mo><mi>p</mi><mo>/</mo><mi>q</mi></mrow></math></span> for some <span><math><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mi>N</mi></mrow></math></span>, then <span><math><mrow><mn>1</mn><mo>/</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>1</mn><mo>/</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⩽</mo><mn>1</mn><mo>/</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>1</mn><mo>/</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>. However, the same conclusion fails for every <span><math><mrow><mi>Υ</mi><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow><mo>⩾</mo><mn>4</mn></mrow></math></span>. Next, we study stepwise underapproximation by <span><math><mi>G</mi></math></span>. Let <span><math><mrow><msub><mrow><mi>e</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mi>θ</mi><mo>−</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mn>1</mn><mo>/</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> be the <sp","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139552045","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":"Large deviation principle of multiplicative Ising models on Markov–Cayley trees","authors":"Jung-Chao Ban ,&nbsp;Wen-Guei Hu ,&nbsp;Zongfan Zhang","doi":"10.1016/j.indag.2024.03.005","DOIUrl":"10.1016/j.indag.2024.03.005","url":null,"abstract":"<div><p>In this paper, we study the large deviation principle (LDP) for two types (Type I and Type II) of multiplicative Ising models. For Types I and II, the explicit formulas for the free energy functions and the associated rate functions are derived. Furthermore, we prove that those free energy functions are differentiable, which indicates that both systems are characterized by a lack of phase transition phenomena.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140150258","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}