Expositiones Mathematicae最新文献

{"title":"Rings of tautological forms on moduli spaces of curves","authors":"Robin de Jong, Stefan van der Lugt","doi":"10.1016/j.exmath.2023.02.008","DOIUrl":"https://doi.org/10.1016/j.exmath.2023.02.008","url":null,"abstract":"","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"54342561","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":"Krull-Remak-Schmidt decompositions in Hom-finite additive categories","authors":"Amit Shah","doi":"10.1016/j.exmath.2022.12.003","DOIUrl":"10.1016/j.exmath.2022.12.003","url":null,"abstract":"<div><p>An additive category in which each object has a Krull-Remak-Schmidt decomposition—that is, a finite direct sum decomposition consisting of objects with local endomorphism rings—is known as a Krull-Schmidt category. A <span><math><mo>Hom</mo></math></span>-finite category is an additive category <span><math><mi>A</mi></math></span> for which there is a commutative unital ring <span><math><mi>k</mi></math></span>, such that each <span><math><mo>Hom</mo></math></span>-set in <span><math><mi>A</mi></math></span> is a finite length <span><math><mi>k</mi></math></span>-module. The aim of this note is to provide a proof that a <span><math><mo>Hom</mo></math></span>-finite category is Krull-Schmidt, if and only if it has split idempotents, if and only if each indecomposable object has a local endomorphism ring.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 1","pages":"Pages 220-237"},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41496827","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 theory of composites perspective on matrix valued Stieltjes functions","authors":"Graeme W. Milton ,&nbsp;Mihai Putinar","doi":"10.1016/j.exmath.2022.12.005","DOIUrl":"10.1016/j.exmath.2022.12.005","url":null,"abstract":"<div><p>A series of physically motivated operations appearing in the study of composite materials are interpreted in terms of elementary continued fraction transforms of matrix valued, rational Stieltjes functions.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 1","pages":"Pages 186-201"},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47108630","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":"Splitting fields of Xn−","authors":"Chandrashekhar B. Khare, Alfio Fabio La Rosa, Gabor Wiese","doi":"10.1016/j.exmath.2023.02.007","DOIUrl":"https://doi.org/10.1016/j.exmath.2023.02.007","url":null,"abstract":"","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45757313","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":"Noncommutative Ck functions and Fréchet derivatives of operator functions","authors":"Evangelos A. Nikitopoulos","doi":"10.1016/j.exmath.2022.12.004","DOIUrl":"https://doi.org/10.1016/j.exmath.2022.12.004","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Fix a unital &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;-algebra &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, and write &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;sa&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; for the set of self-adjoint elements of &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Also, if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is a continuous function, then write &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;sa&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; for the &lt;em&gt;operator function&lt;/em&gt; &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;↦&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; defined via functional calculus. In this paper, we introduce and study a space &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&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; of &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; functions &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; such that, no matter the choice of &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, the operator function &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;sa&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-times continuously Fréchet differentiable. In other words, if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&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;, then &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; “lifts” to a &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; map &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;sa&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, for any (possibly noncommutative) unital &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;-algebra &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. For this reason, we call &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&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; the space of &lt;em&gt;noncommutative&lt;/em&gt; &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; &lt;em&gt;functions&lt;/em&gt;. Our proof that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;sa&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, which requires only knowledge of the Fréchet derivatives of polynomials and operator norm estim","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 1","pages":"Pages 115-163"},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49834269","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 on X0+(","authors":"V. Arul, J. Müller","doi":"10.1016/j.exmath.2023.02.009","DOIUrl":"https://doi.org/10.1016/j.exmath.2023.02.009","url":null,"abstract":"","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46963822","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":"Approximations of the Riley slice","authors":"Alex Elzenaar ,&nbsp;Gaven Martin ,&nbsp;Jeroen Schillewaert","doi":"10.1016/j.exmath.2022.12.002","DOIUrl":"10.1016/j.exmath.2022.12.002","url":null,"abstract":"<div><p>Adapting the ideas of L. Keen and C. Series used in their study of the Riley slice of Schottky groups generated by two parabolics, we explicitly identify ‘half-space’ neighbourhoods of pleating rays which lie completely in the Riley slice. This gives a provable method to determine if a point is in the Riley slice or not. We also discuss the family of Farey polynomials which determine the rational pleating rays and their root set which determines the Riley slice; this leads to a dynamical systems interpretation of the slice. Adapting these methods to the case of Schottky groups generated by two elliptic elements in subsequent work facilitates the programme to identify all the finitely many arithmetic generalised triangle groups and their kin.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 1","pages":"Pages 20-54"},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42835286","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":"Thresholds for the monochromatic clique transversal game","authors":"Csilla Bujtás ,&nbsp;Pakanun Dokyeesun ,&nbsp;Sandi Klavžar","doi":"10.1016/j.exmath.2022.11.001","DOIUrl":"10.1016/j.exmath.2022.11.001","url":null,"abstract":"<div><p>We study a recently introduced two-person combinatorial game, the <span><math><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></math></span>-monochromatic clique transversal game which is played by Alice and Bob on a graph <span><math><mi>G</mi></math></span>. As we observe, this game is equivalent to the <span><math><mrow><mo>(</mo><mi>b</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></math></span>-biased Maker–Breaker game played on the clique-hypergraph of <span><math><mi>G</mi></math></span>. Our main results concern the threshold bias <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> that is the smallest integer <span><math><mi>a</mi></math></span> such that Alice can win in the <span><math><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span>-monochromatic clique transversal game on <span><math><mi>G</mi></math></span> if she is the first to play. Among other results, we determine the possible values of <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> for the disjoint union of graphs, prove a formula for <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> if <span><math><mi>G</mi></math></span> is triangle-free, and obtain the exact values of <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mspace></mspace><mo>□</mo><mspace></mspace><msub><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mspace></mspace><mo>□</mo><mspace></mspace><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, and <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mspace></mspace><mo>□</mo><mspace></mspace><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> for all possible pairs <span><math><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo></mrow></math></span>.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 1","pages":"Pages 202-219"},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43977444","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":"The product of lattice covolume and discrete series formal dimension: p-adic GL(2)","authors":"L.C. Ruth","doi":"10.1016/j.exmath.2022.09.001","DOIUrl":"https://doi.org/10.1016/j.exmath.2022.09.001","url":null,"abstract":"<div><p>Let <span><math><mi>F</mi></math></span> be a nonarchimedean local field of characteristic 0 and residue field of order not divisible by 2. We show how to calculate the product of the covolume of a torsion-free lattice in <span><math><mrow><mi>P</mi><mi>G</mi><mi>L</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> and the formal dimension of a discrete series representation of <span><math><mrow><mi>G</mi><mi>L</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span>. The covolume comes from a theorem of Ihara, and the formal dimensions are contained in results of Corwin, Moy, and Sally. By a theorem going back to Atiyah, and by triviality of the second cohomology group of a free group, the resulting product is the von Neumann dimension of a discrete series representation considered as a representation of a free group factor.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 1","pages":"Pages 55-70"},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49877666","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 analogue of Furstenberg–Sárközy’s theorem and an alternative solution to Waring’s problem over finite fields","authors":"Yeşi̇m Demi̇roğlu Karabulut","doi":"10.1016/j.exmath.2022.10.003","DOIUrl":"https://doi.org/10.1016/j.exmath.2022.10.003","url":null,"abstract":"<div><p>In this paper, we use Cayley digraphs to obtain some new self-contained proofs for Waring’s problem over finite fields, proving that any element of a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> can be written as a sum of <span><math><mi>m</mi></math></span> many <span><math><mrow><mi>k</mi><mtext>th</mtext></mrow></math></span> powers as long as <span><math><mrow><mi>q</mi><mo>&gt;</mo><msup><mrow><mi>k</mi></mrow><mrow><mfrac><mrow><mn>2</mn><mi>m</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup></mrow></math></span>; and we also compute the smallest positive integers <span><math><mi>m</mi></math></span> such that every element of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> can be written as a sum of <span><math><mi>m</mi></math></span> many <span><math><mrow><mi>k</mi><mtext>th</mtext></mrow></math></span> powers for all <span><math><mi>q</mi></math></span> too small to be covered by the above mentioned results when <span><math><mrow><mn>2</mn><mo>⩽</mo><mi>k</mi><mo>⩽</mo><mn>37</mn></mrow></math></span>.</p><p>In the process of developing the proofs mentioned above, we arrive at another result (providing a finite field analogue of Furstenberg–Sárközy’s Theorem) showing that any subset <span><math><mi>E</mi></math></span> of a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> for which <span><math><mrow><mrow><mo>|</mo><mi>E</mi><mo>|</mo></mrow><mo>&gt;</mo><mfrac><mrow><mi>q</mi><mi>k</mi></mrow><mrow><msqrt><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msqrt></mrow></mfrac></mrow></math></span> must contain at least two distinct elements whose difference is a <span><math><mrow><mi>k</mi><mtext>th</mtext></mrow></math></span> power.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 1","pages":"Pages 164-185"},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49877667","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}