MathematikaPub Date : 2023-06-06DOI: 10.1112/mtk.12210
Gautam Aishwarya, Irfan Alam, Dongbin Li, Sergii Myroshnychenko, Oscar Zatarain-Vera
{"title":"Entropic exercises around the Kneser–Poulsen conjecture","authors":"Gautam Aishwarya, Irfan Alam, Dongbin Li, Sergii Myroshnychenko, Oscar Zatarain-Vera","doi":"10.1112/mtk.12210","DOIUrl":"10.1112/mtk.12210","url":null,"abstract":"<p>We develop an information-theoretic approach to study the Kneser–Poulsen conjecture in discrete geometry. This leads us to a broad question regarding whether Rényi entropies of independent sums decrease when one of the summands is contracted by a 1-Lipschitz map. We answer this question affirmatively in various cases.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 3","pages":"841-866"},"PeriodicalIF":0.8,"publicationDate":"2023-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12210","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49598006","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}
MathematikaPub Date : 2023-05-31DOI: 10.1112/mtk.12207
Manami Roy, Ralf Schmidt, Shaoyun Yi
{"title":"Dimension formulas for Siegel modular forms of level 4","authors":"Manami Roy, Ralf Schmidt, Shaoyun Yi","doi":"10.1112/mtk.12207","DOIUrl":"10.1112/mtk.12207","url":null,"abstract":"<p>We prove several dimension formulas for spaces of scalar-valued Siegel modular forms of degree 2 with respect to certain congruence subgroups of level 4. In case of cusp forms, all modular forms considered originate from cuspidal automorphic representations of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>GSp</mi>\u0000 <mo>(</mo>\u0000 <mn>4</mn>\u0000 <mo>,</mo>\u0000 <mi>A</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${rm GSp}(4,{mathbb {A}})$</annotation>\u0000 </semantics></math> whose local component at <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$p=2$</annotation>\u0000 </semantics></math> admits nonzero fixed vectors under the principal congruence subgroup of level 2. Using known dimension formulas combined with dimensions of spaces of fixed vectors in local representations at <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$p=2$</annotation>\u0000 </semantics></math>, we obtain formulas for the number of relevant automorphic representations. These, in turn, lead to new dimension formulas, in particular for Siegel modular forms with respect to the Klingen congruence subgroup of level 4.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 3","pages":"795-840"},"PeriodicalIF":0.8,"publicationDate":"2023-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44248425","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}
MathematikaPub Date : 2023-05-30DOI: 10.1112/mtk.12206
Hung M. Bui, Richard R. Hall
{"title":"A note on the zeros of the derivatives of Hardy's function \u0000 \u0000 \u0000 Z\u0000 (\u0000 t\u0000 )\u0000 \u0000 $Z(t)$","authors":"Hung M. Bui, Richard R. Hall","doi":"10.1112/mtk.12206","DOIUrl":"10.1112/mtk.12206","url":null,"abstract":"<p>Using the twisted fourth moment of the Riemann zeta-function, we study large gaps between consecutive zeros of the derivatives of Hardy's function <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Z</mi>\u0000 <mo>(</mo>\u0000 <mi>t</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$Z(t)$</annotation>\u0000 </semantics></math>, improving upon previous results of Conrey and Ghosh (J. Lond. Math. Soc. <b>32</b> (1985) 193–202), and of the second named author (Acta Arith. 111 (2004) 125–140). We also exhibit small distances between the zeros of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Z</mi>\u0000 <mo>(</mo>\u0000 <mi>t</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$Z(t)$</annotation>\u0000 </semantics></math> and the zeros of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>Z</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mn>2</mn>\u0000 <mi>k</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>t</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$Z^{(2k)}(t)$</annotation>\u0000 </semantics></math> for every <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 <mo>∈</mo>\u0000 <mi>N</mi>\u0000 </mrow>\u0000 <annotation>$kin mathbb {N}$</annotation>\u0000 </semantics></math>, in support of our numerical observation that the zeros of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>Z</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>k</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>t</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$Z^{(k)}(t)$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>Z</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>ℓ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 3","pages":"780-794"},"PeriodicalIF":0.8,"publicationDate":"2023-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12206","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46144088","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}