Research in the Mathematical Sciences最新文献

{"title":"Metric uniform distribution on analytic curves: With an Appendix by Jim Wright.","authors":"Vitaly Bergelson, Joel Moreira","doi":"10.1007/s40687-026-00647-6","DOIUrl":"https://doi.org/10.1007/s40687-026-00647-6","url":null,"abstract":"<p><p>We obtain multidimensional metric uniform distribution results involving sequences in <math> <msup><mrow><mi>R</mi></mrow> <mi>k</mi></msup> </math> parametrized by analytic curves. Our theorems extend the classical theorems of Weyl and Koksma in a variety of ways. One of our main results implies that for any injective sequences <math> <mrow><msub><mi>a</mi> <mn>1</mn></msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub><mi>a</mi> <mi>k</mi></msub> <mo>:</mo> <mi>N</mi> <mo>→</mo> <mi>Z</mi></mrow> </math> the set <dispformula> <math> <mrow> <mtable> <mtr> <mtd> <mrow><mrow><mo>{</mo></mrow> <mrow><mo>(</mo> <msub><mi>x</mi> <mn>1</mn></msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub><mi>x</mi> <mi>k</mi></msub> <mo>)</mo></mrow> <mo>∈</mo> <msup><mrow><mi>R</mi></mrow> <mi>k</mi></msup> <mo>:</mo> <mrow><mo>(</mo></mrow> <msub><mi>a</mi> <mn>1</mn></msub> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> <msub><mi>x</mi> <mn>1</mn></msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub><mi>a</mi> <mi>k</mi></msub> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> <msub><mi>x</mi> <mi>k</mi></msub> <msub><mrow><mo>)</mo></mrow> <mrow><mi>n</mi> <mo>∈</mo> <mi>N</mi></mrow> </msub> <mspace></mspace> <mtext>is uniformly distributed in</mtext> <mspace></mspace> <msup><mrow><mi>T</mi></mrow> <mi>k</mi></msup> <mrow><mo>}</mo></mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </math> </dispformula> has full Lebesgue measure inside any non-degenerate analytic curve <math><mrow><mi>γ</mi> <mo>⊂</mo> <msup><mrow><mi>R</mi></mrow> <mi>k</mi></msup> </mrow> </math> .</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"13 3","pages":"77"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13577984/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148992020","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">On <i>p</i>-adic <i>L</i>-functions for symplectic representations of <ns0:math><ns0:mrow><ns0:mtext>GL</ns0:mtext> <ns0:mo>(</ns0:mo> <ns0:mi>N</ns0:mi> <ns0:mo>)</ns0:mo></ns0:mrow> </ns0:math> over number fields.","authors":"Chris Williams","doi":"10.1007/s40687-026-00635-w","DOIUrl":"10.1007/s40687-026-00635-w","url":null,"abstract":"<p><p>Let <i>F</i> be a number field and <math><mi>π</mi></math> a regular algebraic cuspidal automorphic representation of <math> <mrow><msub><mtext>GL</mtext> <mi>N</mi></msub> <mrow><mo>(</mo> <msub><mi>A</mi> <mi>F</mi></msub> <mo>)</mo></mrow> </mrow> </math> of symplectic type. When <math><mi>π</mi></math> is spherical at all primes <math><mrow><mi>p</mi> <mo>|</mo> <mi>p</mi></mrow> </math> , we construct a <i>p</i>-adic <i>L</i>-function attached to any regular non-critical spin <i>p</i>-refinement <math><mover><mi>π</mi> <mo>~</mo></mover> </math> of <math><mi>π</mi></math> to <i>Q</i>-parahoric level, where <i>Q</i> is the (<i>n</i>, <i>n</i>)-parabolic. More precisely, we construct a distribution <math> <mrow><msub><mi>L</mi> <mi>p</mi></msub> <mrow><mo>(</mo> <mover><mi>π</mi> <mo>~</mo></mover> <mo>)</mo></mrow> </mrow> </math> on the Galois group <math><msub><mtext>Gal</mtext> <mi>p</mi></msub> </math> of the maximal abelian extension of <i>F</i> unramified outside <math><mrow><mi>p</mi> <mi>∞</mi></mrow> </math> and show that it interpolates all the standard critical <i>L</i>-values of <math><mi>π</mi></math> at <i>p</i> (including, for example, cyclotomic and anticyclotomic variation when <i>F</i> is imaginary quadratic). We show that <math> <mrow><msub><mi>L</mi> <mi>p</mi></msub> <mrow><mo>(</mo> <mover><mi>π</mi> <mo>~</mo></mover> <mo>)</mo></mrow> </mrow> </math> satisfies a natural growth condition; in particular, when <math><mover><mi>π</mi> <mo>~</mo></mover> </math> is ordinary, <math> <mrow><msub><mi>L</mi> <mi>p</mi></msub> <mrow><mo>(</mo> <mover><mi>π</mi> <mo>~</mo></mover> <mo>)</mo></mrow> </mrow> </math> is a (bounded) measure on <math><msub><mtext>Gal</mtext> <mi>p</mi></msub> </math> . As a corollary, when <math><mi>π</mi></math> is unitary, has very regular weight, and is <i>Q</i>-ordinary at all <math><mrow><mi>p</mi> <mo>|</mo> <mi>p</mi></mrow> </math> , we deduce non-vanishing <math><mrow><mi>L</mi> <mo>(</mo> <mi>π</mi> <mo>×</mo> <mrow><mo>(</mo> <mi>χ</mi> <mo>∘</mo> <msub><mi>N</mi> <mrow><mi>F</mi> <mo>/</mo> <mi>Q</mi></mrow> </msub> <mo>)</mo></mrow> <mo>,</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> <mo>)</mo> <mo>≠</mo> <mn>0</mn></mrow> </math> of the twisted central value for all but finitely many Dirichlet characters <math><mi>χ</mi></math> of <i>p</i>-power conductor.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"13 2","pages":"52"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13201329/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148043844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"False and partial Eisenstein-type series related to unimodal sequences.","authors":"Kathrin Bringmann, Badri Vishal Pandey, Jan-Willem van Ittersum","doi":"10.1007/s40687-026-00609-y","DOIUrl":"https://doi.org/10.1007/s40687-026-00609-y","url":null,"abstract":"<p><p>Motivated by the fact that the classical Jacobi theta function <math><mi>ϑ</mi></math> is the exponential generating function of the Eisenstein series, we study the exponential Taylor coefficients (in the elliptic variable) of a related natural partial theta function, as well as a false theta function corresponding to the Dedekind eta function. We prove that the space spanned by these objects is closed under differentiation, analogous to the space of quasimodular forms, and that it contains the quasimodular forms themselves. We further provide their Fourier expansions, establish quasimodular completions, and derive a recursive formula for the Taylor coefficients of the logarithm of the unimodal rank generating function, expressed as partition traces of the false and partial objects.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"13 2","pages":"46"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13149652/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147870134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A conjecture of Warnaar-Zudilin from deformations of lie superalgebras.","authors":"Thomas Creutzig, Niklas Garner","doi":"10.1007/s40687-026-00613-2","DOIUrl":"10.1007/s40687-026-00613-2","url":null,"abstract":"<p><p>We prove a collection of <i>q</i>-series identities conjectured by Warnaar and Zudilin and appearing in recent work with H. Kim in the context of superconformal field theory. Our proof utilizes a deformation of the simple affine vertex operator superalgebra <math> <mrow><msub><mi>L</mi> <mi>k</mi></msub> <mrow><mo>(</mo> <msub><mi>osp</mi> <mrow><mn>1</mn> <mo>|</mo> <mn>2</mn> <mi>n</mi></mrow> </msub> <mo>)</mo></mrow> </mrow> </math> into the principal subsuperspace of <math> <mrow><msub><mi>L</mi> <mi>k</mi></msub> <mrow><mo>(</mo> <mi>s</mi> <msub><mi>l</mi> <mrow><mn>1</mn> <mo>|</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn></mrow> </msub> <mo>)</mo></mrow> </mrow> </math> in a manner analogous to earlier work of Feigin-Stoyanovsky. This result fills a gap left by Stoyanovsky, showing that for all positive integers <i>N</i>, <i>k</i> the character of the principal subspace of type <math><msub><mi>A</mi> <mi>N</mi></msub> </math> at level <i>k</i> can be identified with the (super)character of a simple affine vertex operator (super)algebra at the same level.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"13 2","pages":"33"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13046567/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147624547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Applications of dimension interpolation to orthogonal projections.","authors":"Jonathan M Fraser","doi":"10.1007/s40687-025-00496-9","DOIUrl":"https://doi.org/10.1007/s40687-025-00496-9","url":null,"abstract":"<p><p>Dimension interpolation is a novel programme of research which attempts to unify the study of fractal dimension by considering various spectra which live in between well-studied notions of dimension such as Hausdorff, box, Assouad and Fourier dimension. These spectra often reveal novel features not witnessed by the individual notions and this information has applications in many directions. In this survey article, we discuss dimension interpolation broadly and then focus on applications to the dimension theory of orthogonal projections. We focus on three distinct applications coming from three different dimension spectra, namely, the Fourier spectrum, the intermediate dimensions, and the Assouad spectrum. The celebrated Marstrand-Mattila projection theorem gives the Hausdorff dimension of the orthogonal projection of a Borel set in Euclidean space for almost all orthogonal projections. This result has inspired much further research on the dimension theory of projections including the consideration of dimensions other than the Hausdorff dimension, and the study of the exceptional set in the Marstrand-Mattila theorem.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"12 1","pages":"10"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11890391/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143598044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal.","authors":"Alex Fink, Jeffrey Giansiracusa, Noah Giansiracusa, Joshua Mundinger","doi":"10.1007/s40687-025-00517-7","DOIUrl":"https://doi.org/10.1007/s40687-025-00517-7","url":null,"abstract":"<p><p>A \"tropical ideal\" is an ideal in the idempotent semiring of tropical polynomials that is also, degree by degree, a tropical linear space. We introduce a construction based on transversal matroids that canonically extends any principal ideal to a tropical ideal. We call this the Macaulay tropical ideal. It has a universal property: any other extension of the given principal ideal to a tropical ideal with the expected Hilbert function is a weak image of the Macaulay tropical ideal. For each <math><mrow><mi>n</mi> <mo>≥</mo> <mn>2</mn></mrow> </math> and <math><mrow><mi>d</mi> <mo>≥</mo> <mn>1</mn></mrow> </math> , our construction yields a non-realizable degree <i>d</i> hypersurface scheme in <math> <msup><mrow><mi>P</mi></mrow> <mi>n</mi></msup> </math> . Maclagan-Rincón produced a non-realizable line in <math> <msup><mrow><mi>P</mi></mrow> <mi>n</mi></msup> </math> for each <i>n</i>, and for <math><mrow><mo>(</mo> <mi>d</mi> <mo>,</mo> <mi>n</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>)</mo></mrow> </math> the two constructions agree. An appendix by Mundinger compares the Macaulay construction with another method for canonically extending ideals to tropical ideals.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"12 2","pages":"30"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12031988/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144035312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Approximate incidence geometry in the plane.","authors":"Tuomas Orponen","doi":"10.1007/s40687-025-00552-4","DOIUrl":"https://doi.org/10.1007/s40687-025-00552-4","url":null,"abstract":"<p><p>These are lecture notes for a mini-course given in Banff in June 2024. They discuss the problem of bounding the number of <math><mi>δ</mi></math> <i>-incidences</i> <math> <mrow><msub><mi>I</mi> <mi>δ</mi></msub> <mrow><mo>(</mo> <mi>P</mi> <mo>,</mo> <mi>L</mi> <mo>)</mo></mrow> <mo>:</mo> <mo>=</mo> <mrow><mo>{</mo> <mrow><mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>ℓ</mi> <mo>)</mo></mrow> <mo>∈</mo> <mi>P</mi> <mo>×</mo> <mi>L</mi> <mo>:</mo> <mi>p</mi> <mo>∈</mo> <msub><mrow><mo>[</mo> <mi>ℓ</mi> <mo>]</mo></mrow> <mi>δ</mi></msub> <mo>}</mo></mrow> </mrow> </math> under various hypotheses on <math><mrow><mi>P</mi> <mo>⊂</mo> <msup><mrow><mi>R</mi></mrow> <mn>2</mn></msup> </mrow> </math> and <math><mrow><mi>L</mi> <mo>⊂</mo> <mi>A</mi> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo></mrow> </math> . The main focus will be on hypotheses relevant for the <i>Furstenberg set problem</i>.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"12 4","pages":"65"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12413431/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145015229","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Residue class biases in unrestricted partitions, partitions into distinct parts, and overpartitions.","authors":"Michael J Schlosser, Nian Hong Zhou","doi":"10.1007/s40687-025-00502-0","DOIUrl":"https://doi.org/10.1007/s40687-025-00502-0","url":null,"abstract":"<p><p>We prove specific biases in the number of occurrences of parts belonging to two different residue classes <i>a</i> and <i>b</i>, modulo a fixed nonnegative integer <i>m</i>, for the sets of unrestricted partitions, partitions into distinct parts, and overpartitions. These biases follow from inequalities for residue-weighted partition functions for the respective sets of partitions. We also establish asymptotic formulas for the numbers of partitions of size <i>n</i> that belong to these sets of partitions and have a symmetric residue class bias (i.e., for <math><mrow><mn>1</mn> <mo>≤</mo> <mi>a</mi> <mo><</mo> <mi>m</mi> <mo>/</mo> <mn>2</mn></mrow> </math> and <math><mrow><mi>b</mi> <mo>=</mo> <mi>m</mi> <mo>-</mo> <mi>a</mi></mrow> </math> ), as <i>n</i> tends to infinity.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"12 1","pages":"17"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11845404/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143484297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Proceedings of the 17th International Workshop on Real and Complex Singularities","authors":"Raimundo Nonato Araújo dos Santos, Alex Carlucci Rezende, Toru Ohmoto, Kentaro Saji","doi":"10.1007/s40687-024-00465-8","DOIUrl":"https://doi.org/10.1007/s40687-024-00465-8","url":null,"abstract":"","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"293 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Splitting hypergeometric functions over roots of unity","authors":"Dermot McCarthy, Mohit Tripathi","doi":"10.1007/s40687-024-00468-5","DOIUrl":"https://doi.org/10.1007/s40687-024-00468-5","url":null,"abstract":"<p>We examine hypergeometric functions in the finite field, <i>p</i>-adic and classical settings. In each setting, we prove a formula which splits the hypergeometric function into a sum of lower order functions whose arguments differ by roots of unity. We provide multiple applications of these results, including new reduction and summation formulas for finite field hypergeometric functions, along with classical analogues; evaluations of special values of these functions which apply in both the finite field and <i>p</i>-adic settings; and new relations to Fourier coefficients of modular forms.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"6 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217530","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}