Mathematika最新文献

{"title":"Zeros of dirichlet L-functions near the critical line","authors":"George Dickinson","doi":"10.1112/mtk.12239","DOIUrl":"https://doi.org/10.1112/mtk.12239","url":null,"abstract":"<p>We prove an upper bound on the density of zeros very close to the critical line of the family of Dirichlet <i>L</i>-functions of modulus <i>q</i> at height <i>T</i>. To do this, we derive an asymptotic for the twisted second moment of Dirichlet <i>L</i>-functions uniformly in <i>q</i> and <i>t</i>. As a second application of the asymptotic formula, we prove that, for every integer <i>q</i>, at least 38.2% of zeros of the primitive Dirichlet <i>L</i>-functions of modulus <i>q</i> lie on the critical line.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12239","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139109892","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":"Discrepancy of arithmetic progressions in grids","authors":"Jacob Fox,&nbsp;Max Wenqiang Xu,&nbsp;Yunkun Zhou","doi":"10.1112/mtk.12237","DOIUrl":"https://doi.org/10.1112/mtk.12237","url":null,"abstract":"<p>We prove that the discrepancy of arithmetic progressions in the <i>d</i>-dimensional grid <math>\u0000 <semantics>\u0000 <msup>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mi>⋯</mi>\u0000 <mo>,</mo>\u0000 <mi>N</mi>\u0000 <mo>}</mo>\u0000 </mrow>\u0000 <mi>d</mi>\u0000 </msup>\u0000 <annotation>$lbrace 1, dots, Nrbrace ^d$</annotation>\u0000 </semantics></math> is within a constant factor depending only on <i>d</i> of <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>N</mi>\u0000 <mfrac>\u0000 <mi>d</mi>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>d</mi>\u0000 <mo>+</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </mfrac>\u0000 </msup>\u0000 <annotation>$N^{frac{d}{2d+2}}$</annotation>\u0000 </semantics></math>. This extends the case <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>=</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$d=1$</annotation>\u0000 </semantics></math>, which is a celebrated result of Roth and of Matoušek and Spencer, and removes the polylogarithmic factor from the previous upper bound of Valkó from about two decades ago. We further prove similarly tight bounds for grids of differing side lengths in many cases.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139090685","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":"A quantitative Hasse principle for weighted quartic forms","authors":"Daniel Flores","doi":"10.1112/mtk.12236","DOIUrl":"https://doi.org/10.1112/mtk.12236","url":null,"abstract":"<p>We derive, via the Hardy–Littlewood method, an asymptotic formula for the number of integral zeros of a particular class of weighted quartic forms under the assumption of nonsingular local solubility. Our polynomials <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>x</mi>\u0000 <mo>,</mo>\u0000 <mi>y</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>∈</mo>\u0000 <mi>Z</mi>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <msub>\u0000 <mi>x</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mtext>…</mtext>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>x</mi>\u0000 <msub>\u0000 <mi>s</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>y</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mtext>…</mtext>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>y</mi>\u0000 <msub>\u0000 <mi>s</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 </msub>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$F({mathbf {x}},{mathbf {y}}) in {mathbb {Z}}[x_1,ldots ,x_{s_1},y_1,ldots ,y_{s_2}]$</annotation>\u0000 </semantics></math> satisfy the condition that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>λ</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mi>x</mi>\u0000 <mo>,</mo>\u0000 <mi>λ</mi>\u0000 <mi>y</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>=</mo>\u0000 <msup>\u0000 <mi>λ</mi>\u0000 <mn>4</mn>\u0000 </msup>\u0000 <mi>F</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>x</mi>\u0000 <mo>,</mo>\u0000 <mi>y</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$F(lambda ^2 {mathbf {x}}, lambda {mathbf {y}}) = lambda ^4 F({mathbf {x}},{mathbf {y}})$</annotation>\u0000 </semantics></math>. Our conclusions improve on those that would follow from a direct application of the methods of Birch. For example, we show that in m","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139047282","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":"A unified approach to higher order discrete and smooth isoperimetric inequalities","authors":"Kwok-Kun Kwong","doi":"10.1112/mtk.12238","DOIUrl":"10.1112/mtk.12238","url":null,"abstract":"<p>We present a unified approach to derive sharp isoperimetric-type inequalities of arbitrary high order. In particular, we obtain (i) sharp high-order discrete polygonal isoperimetric-type inequalities, (ii) sharp high-order isoperimetric-type inequalities for smooth curves with both upper and lower bounds for the isoperimetric deficit, and (iii) sharp higher order Chernoff-type inequalities involving a generalized width function and higher order locus of curvature centers. Our approach involves obtaining higher order discrete or smooth Wirtinger inequalities via Fourier analysis, by examining a family of linear operators. The key to our approach is identifying the appropriate linear operator and translating the analytic inequalities into geometric ones.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12238","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138822576","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":"On the homotopy type of multipath complexes","authors":"Luigi Caputi,&nbsp;Carlo Collari,&nbsp;Sabino Di Trani,&nbsp;Jason P. Smith","doi":"10.1112/mtk.12235","DOIUrl":"https://doi.org/10.1112/mtk.12235","url":null,"abstract":"<p>A multipath in a directed graph is a disjoint union of paths. The multipath complex of a directed graph <math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>${tt G}$</annotation>\u0000 </semantics></math> is the simplicial complex whose faces are the multipaths of <math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>${tt G}$</annotation>\u0000 </semantics></math>. We compute Euler characteristics, and associated generating functions, of the multipath complexes of directed graphs from certain families, including transitive tournaments and complete bipartite graphs. We show that if <math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>${tt G}$</annotation>\u0000 </semantics></math> is a linear graph, polygon, small grid or transitive tournament, then the homotopy type of the multipath complex of <math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>${tt G}$</annotation>\u0000 </semantics></math> is always contractible or a wedge of spheres. We introduce a new technique for decomposing directed graphs into dynamical regions, which allows us to simplify the homotopy computations.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138558249","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":"Kloosterman sums do not correlate with periodic functions","authors":"Raphael S. Steiner","doi":"10.1112/mtk.12232","DOIUrl":"https://doi.org/10.1112/mtk.12232","url":null,"abstract":"<p>We provide uniform bounds for sums of Kloosterman sums in <i>all</i> arithmetic progressions. As a consequence, we find that Kloosterman sums do not correlate with periodic functions.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138155339","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":"A decoupling interpretation of an old argument for Vinogradov's Mean Value Theorem","authors":"Brian Cook,&nbsp;Kevin Hughes,&nbsp;Zane Kun Li,&nbsp;Akshat Mudgal,&nbsp;Olivier Robert,&nbsp;Po-Lam Yung","doi":"10.1112/mtk.12231","DOIUrl":"https://doi.org/10.1112/mtk.12231","url":null,"abstract":"<p>We interpret into decoupling language a refinement of a 1973 argument due to Karatsuba on Vinogradov's mean value theorem. The main goal of our argument is to answer what precisely solution counting in older partial progress on Vinogradov's mean value theorem corresponds to in Fourier decoupling theory.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12231","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"109168015","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":"Quasi-invariance of modulus and quasisymmetry of weakly (L,M)-quasisymmetric maps in metric spaces","authors":"Tao Cheng,&nbsp;Shanshuang Yang","doi":"10.1112/mtk.12233","DOIUrl":"https://doi.org/10.1112/mtk.12233","url":null,"abstract":"<p>This paper contributes to the study of a fundamental problem in the theory of quasiconformal analysis: under what conditions local quasiconformality of a homeomorphism implies its global quasisymmetry. We show that in general metric spaces local regularity and some connectivity together with the Loewner condition are necessary and sufficient for a quasiconformal map to be globally quasisymmetric with respect to the internal metrics. In this endeavor, two major new ingredients are used. One is the recently introduced concept of weakly <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>L</mi>\u0000 <mo>,</mo>\u0000 <mi>M</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(L,M)$</annotation>\u0000 </semantics></math>-quasisymmetry, serving as a bridge between local quasiconformality and global quasisymmetry. Another is the quasi-invariance of conformal modulus under weakly <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>L</mi>\u0000 <mo>,</mo>\u0000 <mi>M</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(L,M)$</annotation>\u0000 </semantics></math>-quasisymmetric maps, which is developed in this paper.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"109168014","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":"Coloring unions of nearly disjoint hypergraph cliques","authors":"Dhruv Mubayi,&nbsp;Jacques Verstraete","doi":"10.1112/mtk.12234","DOIUrl":"10.1112/mtk.12234","url":null,"abstract":"<p>We consider the maximum chromatic number of hypergraphs consisting of cliques that have pairwise small intersections. Designs of the appropriate parameters produce optimal constructions, but these are generally known to exist only when the number of cliques is exponential in the clique size (Glock, Kühn, Lo, and Osthus, <i>Mem. Amer. Math. Soc</i>. <b>284</b> (2023) v+131 pp; Keevash, Preprint; Rödl, <i>Eur. J. Combin</i>. 6 (1985) 69–78). We construct near designs where the number of cliques is polynomial in the clique size, and show that they have large chromatic number. The case when the cliques have pairwise intersections of size at most one seems particularly challenging. Here, we give lower bounds by analyzing a random greedy hypergraph process. We also consider the related question of determining the maximum number of caps in a finite projective/affine plane and obtain nontrivial upper and lower bounds.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135243024","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":"Upper bounds for the constants of Bennett's inequality and the Gale–Berlekamp switching game","authors":"Daniel Pellegrino,&nbsp;Anselmo Raposo Jr.","doi":"10.1112/mtk.12229","DOIUrl":"10.1112/mtk.12229","url":null,"abstract":"<p>In 1977, G. Bennett proved, by means of nondeterministic methods, an inequality that plays a fundamental role in a series of optimization problems. More precisely, Bennett's inequality shows that, for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>p</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>p</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <mo>∈</mo>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mi>∞</mi>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$p_{1},p_{2} in [1,infty ]$</annotation>\u0000 </semantics></math> and all positive integers <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>n</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>n</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$n_{1},n_{2}$</annotation>\u0000 </semantics></math>, there exists a bilinear form <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>A</mi>\u0000 <mrow>\u0000 <msub>\u0000 <mi>n</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>n</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 </msub>\u0000 <mrow>\u0000 <mo>:</mo>\u0000 <mo>(</mo>\u0000 </mrow>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <msub>\u0000 <mi>n</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 </msup>\u0000 <msub>\u0000 <mrow>\u0000 <mo>,</mo>\u0000 <mo>∥</mo>\u0000 <mo>·</mo>\u0000 <mo>∥</mo>\u0000 </mrow>\u0000 <msub>\u0000 <mi>p</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 </msub>\u0000 <mrow>\u0000 <mo>)</mo>\u0000 <mo>×</mo>\u0000 <mo>(</mo>\u0000 </mrow>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <msub>\u0000 <mi>n</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 </msup>\u0000 <msub>\u0000 <mrow>\u0000 <mo>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136234350","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}