Expositiones Mathematicae最新文献

{"title":"On an extension of Fuglede’s theory of pseudo-balayage and its applications","authors":"Natalia Zorii","doi":"10.1016/j.exmath.2024.125630","DOIUrl":"10.1016/j.exmath.2024.125630","url":null,"abstract":"<div><div>For suitable function kernels <span><math><mi>κ</mi></math></span> on a locally compact space, we develop a theory of inner pseudo-balayage <span><math><msup><mrow><mover><mrow><mi>ω</mi></mrow><mrow><mo>̂</mo></mrow></mover></mrow><mrow><mi>A</mi></mrow></msup></math></span> of signed (Radon) measures <span><math><mi>ω</mi></math></span> of finite energy onto a quasiclosed set <span><math><mi>A</mi></math></span>, <span><math><msup><mrow><mover><mrow><mi>ω</mi></mrow><mrow><mo>̂</mo></mrow></mover></mrow><mrow><mi>A</mi></mrow></msup></math></span> being defined as the solution to the problem of minimizing the Gauss functional <span><span><span><math><mrow><mo>∫</mo><mi>κ</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mspace></mspace><mi>d</mi><mrow><mo>(</mo><mi>μ</mi><mo>⊗</mo><mi>μ</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>−</mo><mn>2</mn><mo>∫</mo><mi>κ</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mspace></mspace><mi>d</mi><mrow><mo>(</mo><mi>μ</mi><mo>⊗</mo><mi>ω</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>where <span><math><mi>μ</mi></math></span> ranges over all positive measures of finite energy concentrated on the set <span><math><mi>A</mi></math></span>. If <span><math><mi>A</mi></math></span> is Borel, the concept of inner pseudo-balayage <span><math><msup><mrow><mover><mrow><mi>ω</mi></mrow><mrow><mo>̂</mo></mrow></mover></mrow><mrow><mi>A</mi></mrow></msup></math></span> is shown to coincide with that of outer pseudo-balayage, introduced in Fuglede’s work (Fuglede, 2016), which was however only concerned with <span><math><mrow><mi>ω</mi><mo>⩾</mo><mn>0</mn></mrow></math></span>, whereas the investigation of signed <span><math><mi>ω</mi></math></span> requires essentially different methods and approaches. The theory of pseudo-balayage thereby established enables us to improve substantially our recent results on the well-known inner Gauss variational problem (Zorii, 2024), by strengthening their formulations and/or by extending the area of their validity. This study covers many interesting kernels in classical and modern potential theory, which looks promising for further applications.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125630"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395461","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 complete Bernstein function related to the fractal dimension of Pascal’s pyramid modulo a prime","authors":"Christian Berg","doi":"10.1016/j.exmath.2024.125601","DOIUrl":"10.1016/j.exmath.2024.125601","url":null,"abstract":"<div><div>Let <span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>log</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>r</mi><mi>x</mi><mo>)</mo></mrow><mo>/</mo><mo>log</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>x</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>. We prove that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> is a complete Bernstein function for <span><math><mrow><mn>0</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><mn>1</mn></mrow></math></span> and a Stieltjes function for <span><math><mrow><mn>1</mn><mo>≤</mo><mi>r</mi></mrow></math></span>. This answers a conjecture of David Bradley that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> is a Bernstein function when <span><math><mrow><mn>0</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><mn>1</mn></mrow></math></span>.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125601"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930603","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":"Reprint of: Extremal length and duality","authors":"Kai Rajala","doi":"10.1016/j.exmath.2025.125746","DOIUrl":"10.1016/j.exmath.2025.125746","url":null,"abstract":"<div><div>Classical extremal length (or conformal modulus) is a conformal invariant involving families of paths on the Riemann sphere. In “Extremal length and functional completion”, Fuglede initiated an abstract theory of extremal length which has since been widely applied. Concentrating on duality properties and applications to quasiconformal analysis, we demonstrate the flexibility of the theory and present recent advances in three different settings:</div><div>(1) Extremal length and uniformization of metric surfaces.</div><div>(2) Extremal length of families of surfaces and quasiconformal maps between <span><math><mi>n</mi></math></span>-dimensional spaces.</div><div>(3) Schramm’s transboundary extremal length and conformal maps between multiply connected plane domains.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125746"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395333","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":"Weaving information packets","authors":"Ole Christensen ,&nbsp;Hong Oh Kim ,&nbsp;Rae Young Kim","doi":"10.1016/j.exmath.2025.125720","DOIUrl":"10.1016/j.exmath.2025.125720","url":null,"abstract":"<div><div>The concept of weaving of frames for Hilbert spaces was introduced by Bemrose et al. in 2016. Two frames <span><math><mrow><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>k</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo>,</mo><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>k</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow></math></span> are woven if the “mixed system” <span><math><mrow><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>k</mi><mo>∈</mo><mi>σ</mi></mrow></msub><mo>∪</mo><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>k</mi><mo>∈</mo><mi>I</mi><mo>∖</mo><mi>σ</mi></mrow></msub></mrow></math></span> is a frame for each index set <span><math><mrow><mi>σ</mi><mo>⊂</mo><mi>I</mi><mo>;</mo></mrow></math></span> that is, processing a signal using two woven frames yields a certain stability against loss of information. The concept easily extends to <span><math><mi>N</mi></math></span> frames, for any integer <span><math><mrow><mi>N</mi><mo>&gt;</mo><mn>2</mn><mo>.</mo></mrow></math></span> Unfortunately it is nontrivial to construct useful woven frames, and the literature is sparse concerning explicit constructions. In this paper we introduce so-called information packets, which contain as well frames as fusion frames as special case. The concept of woven frames immediately generalizes to information packets, and we demonstrate how to construct practically relevant woven information packets based on particular wavelet systems in <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>.</mo></mrow></math></span> Interestingly, we show that certain wavelet systems can be split into <span><math><mi>N</mi></math></span> woven information packets, for any integer <span><math><mrow><mi>N</mi><mo>≥</mo><mn>2</mn><mo>.</mo></mrow></math></span> We finally consider corresponding questions for Gabor system in <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span> and prove that for any fixed <span><math><mrow><mi>N</mi><mo>∈</mo><mi>N</mi></mrow></math></span> we can find a Gabor frame that can be split into <span><math><mi>N</mi></math></span> woven information packets; however, in contrast to the wavelet case, the density conditions for Gabor system excludes the possibility of finding a single Gabor frame that works simultaneously for all <span><math><mrow><mi>N</mi><mo>∈</mo><mi>N</mi><mo>.</mo></mrow></math></span></div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125720"},"PeriodicalIF":0.9,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145117694","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":"Galois groups in enumerative geometry and applications","authors":"Frank Sottile ,&nbsp;Thomas Yahl","doi":"10.1016/j.exmath.2025.125731","DOIUrl":"10.1016/j.exmath.2025.125731","url":null,"abstract":"<div><div>As Jordan observed in 1870, just as univariate polynomials have Galois groups, so do problems in enumerative geometry. Despite this pedigree, the study of Galois groups in enumerative geometry was dormant for a century, with a systematic study only occurring in the past 15 years. We discuss the current directions of this study, including open problems and conjectures.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125731"},"PeriodicalIF":0.9,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145324319","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":"Periodic source detection in discrete dynamical systems via space–time sampling","authors":"Akram Aldroubi ,&nbsp;Carlos Cabrelli ,&nbsp;Ursula Molter","doi":"10.1016/j.exmath.2025.125730","DOIUrl":"10.1016/j.exmath.2025.125730","url":null,"abstract":"<div><div>In this paper, we examine a discrete dynamical system defined by <span><math><mrow><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>=</mo><mi>A</mi><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>+</mo><mi>w</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mi>x</mi></math></span> takes values in a Hilbert space <span><math><mi>H</mi></math></span> and <span><math><mi>w</mi></math></span> is a periodic source with values in a fixed closed subspace <span><math><mi>W</mi></math></span> of <span><math><mi>H</mi></math></span>. Our goal is to identify conditions on some spatial sampling system <span><math><mrow><mi>G</mi><mo>=</mo><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msub></mrow></math></span> of <span><math><mi>H</mi></math></span> that enable stable recovery of the unknown source term <span><math><mi>w</mi></math></span> from space–time samples <span><math><msub><mrow><mrow><mo>{</mo><mrow><mo>〈</mo><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>〉</mo></mrow><mo>}</mo></mrow></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn><mo>,</mo><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msub></math></span>. We provide necessary and sufficient conditions on <span><math><mrow><mi>G</mi><mo>=</mo><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msub></mrow></math></span> to ensure stable recovery of any <span><math><mrow><mi>w</mi><mo>∈</mo><mi>W</mi></mrow></math></span>. Additionally, we explicitly construct an operator <span><math><mi>R</mi></math></span>, dependent on <span><math><mi>G</mi></math></span>, such that <span><math><mrow><mi>R</mi><msub><mrow><mrow><mo>{</mo><mrow><mo>〈</mo><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>〉</mo></mrow><mo>}</mo></mrow></mrow><mrow><mi>n</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>=</mo><mi>w</mi></mrow></math></span>.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125730"},"PeriodicalIF":0.9,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145157524","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":"Some examples of affine isometries of Banach spaces arising from 1-D dynamics","authors":"Andrés Navas","doi":"10.1016/j.exmath.2025.125716","DOIUrl":"10.1016/j.exmath.2025.125716","url":null,"abstract":"<div><div>We provide a large family of examples of affine isometries of the Banach spaces <span><math><mrow><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> that are fixed-point-free despite being recurrent (in particular, they have zero drift). These come from natural cocycles on the group of circle diffeomorphisms, namely the logarithmic, affine and (a variation of the) Schwarzian derivative. Quite interestingly, they arise from diffeomorphisms that are generic in an appropriate context. We also show how to promote these examples in order to obtain families of commuting isometries satisfying the same properties.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125716"},"PeriodicalIF":0.9,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144757943","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":"Computation of Lyapunov exponents of matrix products","authors":"Ai Hua Fan ,&nbsp;Evgeny Verbitskiy","doi":"10.1016/j.exmath.2025.125733","DOIUrl":"10.1016/j.exmath.2025.125733","url":null,"abstract":"<div><div>For <span><math><mi>m</mi></math></span> given square matrices <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></math></span>\u0000 (<span><math><mrow><mi>m</mi><mo>≥</mo><mn>2</mn></mrow></math></span>), one of which is assumed to be of rank 1, and for a given sequence <span><math><mrow><mo>(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></math></span> in <span><math><msup><mrow><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow></mrow><mrow><mi>N</mi></mrow></msup></math></span>, the following limit, if it exists, <span><math><mrow><mi>L</mi><mrow><mo>(</mo><mi>ω</mi><mo>)</mo></mrow><mo>≔</mo><munder><mrow><mo>lim</mo></mrow><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></munder><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo>log</mo><mo>‖</mo><msub><mrow><mi>A</mi></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>⋯</mo><msub><mrow><mi>A</mi></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><mo>‖</mo></mrow></math></span> defines the Lyapunov exponent of the sequence of matrices <span><math><msub><mrow><mrow><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span>. It is proven that the Lyapunov exponent <span><math><mrow><mi>L</mi><mrow><mo>(</mo><mi>ω</mi><mo>)</mo></mrow></mrow></math></span> has a closed-form expression under certain conditions. One special case arises when <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>’s are non-negative and <span><math><mi>ω</mi></math></span> is generic with respect to some shift-invariant measure; a second special case occurs when <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>’s (for <span><math><mrow><mn>1</mn><mo>≤</mo><mi>j</mi><mo>&lt;</mo><mi>m</mi></mrow></math></span>) are invertible and <span><math><mi>ω</mi></math></span> is a typical point with respect to some shift-ergodic measure. Substitutive sequences and characteristic sequences of <span><math><mi>B</mi></math></span>-free integers are considered as examples. An application is presented for the computation of multifractal spectrum of weighted Birkhoff averages.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125733"},"PeriodicalIF":0.9,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145361565","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":"Any function I can actually write down is measurable, right?","authors":"James E. Hanson","doi":"10.1016/j.exmath.2025.125718","DOIUrl":"10.1016/j.exmath.2025.125718","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this expository paper aimed at a general mathematical audience, we discuss how to combine certain classic theorems of set-theoretic inner model theory and effective descriptive set theory with work on Hilbert’s tenth problem and universal Diophantine equations to produce the following surprising result: There is a specific polynomial &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;p&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;mi&gt;y&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&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;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;70&lt;/mn&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 degree 7 with integer coefficients such that it is independent of &lt;span&gt;&lt;math&gt;&lt;mi&gt;ZFC&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; (and much stronger theories) whether the function &lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;f&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;mo&gt;=&lt;/mo&gt;&lt;munder&gt;&lt;mrow&gt;&lt;mo&gt;inf&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;munder&gt;&lt;mrow&gt;&lt;mo&gt;sup&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;munder&gt;&lt;mrow&gt;&lt;mo&gt;inf&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;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;munder&gt;&lt;mrow&gt;&lt;mo&gt;sup&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;̄&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;70&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mi&gt;p&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;mi&gt;y&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;̄&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;is Lebesgue measurable. We also give similarly defined &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;g&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;mi&gt;y&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; with the property that the statement “&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;g&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;mi&gt;r&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is measurable for every &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;” has large cardinal consistency strength (and in particular implies the consistency of &lt;span&gt;&lt;math&gt;&lt;mi&gt;ZFC&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;) and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&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;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;16&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; can consistently be the indicator functions of a Banach–Tarski paradoxical decomposition of the sphere.&lt;/div&gt;&lt;div&gt;Finally, we discuss some situations in which measurability of analogously defined functions can be concluded by inspection, which touches on model-theoretic o-minimality and the fact that sufficiently strong large cardinal hypotheses (such as Vopěnka’s principle and much weaker assumptions","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125718"},"PeriodicalIF":0.9,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144851911","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":"Sums of two squares and the tau-function: Ramanujan’s trail","authors":"Bruce C. Berndt ,&nbsp;Pieter Moree","doi":"10.1016/j.exmath.2025.125721","DOIUrl":"10.1016/j.exmath.2025.125721","url":null,"abstract":"<div><div>Ramanujan, in his famous first letter to Hardy, claimed a very precise estimate for the number of integers that can be written as a sum of two squares. Far less well-known is that he also made further claims of a similar nature for the non-divisibility of the Ramanujan tau-function for certain primes. In this survey, we provide more historical details and also discuss related later developments. These show that, as so often, Ramanujan was an explorer in a fascinating wilderness, leaving behind him a beckoning trail.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125721"},"PeriodicalIF":0.9,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145048954","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}