We prove a weighted Sato-Tate law for the Satake parameters of automorphic forms on $rm{GSp}_4$ with respect to a fairly general congruence subgroup $H$ whose level tends to infinity. When the level is squarefree we refine our result to the cuspidal spectrum. The ingredients are the $rm{GSp}_4$ Kuznetsov formula and the explicit calculation of local integrals involved in the Whittaker coefficients of $rm{GSp}_4$ Eisenstein series. We also discuss how the problem of bounding the continuous spectrum in the level aspect naturally leads to some combinatorial questions involving the double cosets in $P backslash G / H$, for each parabolic subgroup $P$.
我们证明了$rm{GSp}_4$上自形形的加权萨托-塔特定律,该定律是关于水平趋于无穷大的一般同余子群$H$的。当水平无平方时,我们将结果细化为尖顶谱。其要素是$rm{GSp}_4$库兹涅佐夫公式和$rm{GSp}_4$爱森斯坦级数的维特克系数所涉及的局部积分的显式计算。我们还讨论了在水平方面约束连续谱的问题如何自然地引出一些组合问题,这些问题涉及每个抛物线子群 $P$ 的 $Pbackslash G / H$ 中的双余弦。
{"title":"A weighted vertical Sato-Tate law for Maaß forms on $rm{GSp}_4$","authors":"Félicien Comtat","doi":"arxiv-2409.06027","DOIUrl":"https://doi.org/arxiv-2409.06027","url":null,"abstract":"We prove a weighted Sato-Tate law for the Satake parameters of automorphic\u0000forms on $rm{GSp}_4$ with respect to a fairly general congruence subgroup $H$\u0000whose level tends to infinity. When the level is squarefree we refine our\u0000result to the cuspidal spectrum. The ingredients are the $rm{GSp}_4$ Kuznetsov\u0000formula and the explicit calculation of local integrals involved in the\u0000Whittaker coefficients of $rm{GSp}_4$ Eisenstein series. We also discuss how\u0000the problem of bounding the continuous spectrum in the level aspect naturally\u0000leads to some combinatorial questions involving the double cosets in $P\u0000backslash G / H$, for each parabolic subgroup $P$.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203835","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The discrepancy of a sequence measures how quickly it approaches a uniform distribution. Given a natural number $d$, any collection of one-dimensional so-called low discrepancy sequences $left{S_i:1le i le dright}$ can be concatenated to create a $d$-dimensional $textit{hybrid sequence}$ $(S_1,dots,S_d)$. Since their introduction by Spanier in 1995, many connections between the discrepancy of a hybrid sequence and the discrepancy of its component sequences have been discovered. However, a proof that a hybrid sequence is capable of being low discrepancy has remained elusive. This paper remedies this by providing an explicit connection between Diophantine approximation over function fields and two dimensional low discrepancy hybrid sequences. Specifically, let $mathbb{F}_q$ be the finite field of cardinality $q$. It is shown that some real numbered hybrid sequence $mathbf{H}(Theta(t),P(t)):=textbf{H}(Theta,P)$ built from the digital Kronecker sequence associated to a Laurent series $Theta(t)inmathbb{F}_q((t^{-1}))$ and the digital Van der Corput sequence associated to an irreducible polynomial $P(t)inmathbb{F}_q[t]$ meets the above property. More precisely, if $Theta(t)$ is a counterexample to the so called $t$$textit{-adic Littlewood Conjecture}$ ($t$-$LC$), then another Laurent series $Phi(t)inmathbb{F}_q((t^{-1}))$ induced from $Theta(t)$ and $P(t)$ can be constructed so that $mathbf{H}(Phi,P)$ is low discrepancy. Such counterexamples to $t$-$LC$ are known over a number of finite fields by, on the one hand, Adiceam, Nesharim and Lunnon, and on the other, by Garrett and the author.
{"title":"Low Discrepancy Digital Kronecker-Van der Corput Sequences","authors":"Steven Robertson","doi":"arxiv-2409.05469","DOIUrl":"https://doi.org/arxiv-2409.05469","url":null,"abstract":"The discrepancy of a sequence measures how quickly it approaches a uniform\u0000distribution. Given a natural number $d$, any collection of one-dimensional\u0000so-called low discrepancy sequences $left{S_i:1le i le dright}$ can be\u0000concatenated to create a $d$-dimensional $textit{hybrid sequence}$\u0000$(S_1,dots,S_d)$. Since their introduction by Spanier in 1995, many\u0000connections between the discrepancy of a hybrid sequence and the discrepancy of\u0000its component sequences have been discovered. However, a proof that a hybrid\u0000sequence is capable of being low discrepancy has remained elusive. This paper\u0000remedies this by providing an explicit connection between Diophantine\u0000approximation over function fields and two dimensional low discrepancy hybrid\u0000sequences. Specifically, let $mathbb{F}_q$ be the finite field of cardinality $q$. It\u0000is shown that some real numbered hybrid sequence\u0000$mathbf{H}(Theta(t),P(t)):=textbf{H}(Theta,P)$ built from the digital\u0000Kronecker sequence associated to a Laurent series\u0000$Theta(t)inmathbb{F}_q((t^{-1}))$ and the digital Van der Corput sequence\u0000associated to an irreducible polynomial $P(t)inmathbb{F}_q[t]$ meets the\u0000above property. More precisely, if $Theta(t)$ is a counterexample to the so\u0000called $t$$textit{-adic Littlewood Conjecture}$ ($t$-$LC$), then another\u0000Laurent series $Phi(t)inmathbb{F}_q((t^{-1}))$ induced from $Theta(t)$ and\u0000$P(t)$ can be constructed so that $mathbf{H}(Phi,P)$ is low discrepancy. Such\u0000counterexamples to $t$-$LC$ are known over a number of finite fields by, on the\u0000one hand, Adiceam, Nesharim and Lunnon, and on the other, by Garrett and the\u0000author.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the property of uniform discreteness within discrete orbits of non-uniform lattices in $SL_2(mathbb{R})$, acting on $mathbb{R}^2$ by linear transformations. We provide a new proof of the conditions under which the orbit of a non-uniform lattice in $SL_2(mathbb{R})$ is uniformly discrete, by using Diophantine properties. Our results include a detailed analysis of the asymptotic behavior of the error terms. Focusing on a specific group $Gamma$ and a discrete orbit of it, $S$, the main result of this paper is that for any $epsilon>0$, three points in $S$ can be found on a horizontal line within distance $epsilon$ of each other. This gives a partial result toward a conjecture of Leli`evre. The set $S$ and group $Gamma$ are respectively the set of long cylinder holonomy vectors, and Veech group, of the "golden L" translation surface.
{"title":"Uniform Discreteness of Discrete Orbits of Non-Uniform Lattices in $SL_2(mathbb{R})$","authors":"Sahar Bashan","doi":"arxiv-2409.05935","DOIUrl":"https://doi.org/arxiv-2409.05935","url":null,"abstract":"We study the property of uniform discreteness within discrete orbits of\u0000non-uniform lattices in $SL_2(mathbb{R})$, acting on $mathbb{R}^2$ by linear\u0000transformations. We provide a new proof of the conditions under which the orbit\u0000of a non-uniform lattice in $SL_2(mathbb{R})$ is uniformly discrete, by using\u0000Diophantine properties. Our results include a detailed analysis of the\u0000asymptotic behavior of the error terms. Focusing on a specific group $Gamma$\u0000and a discrete orbit of it, $S$, the main result of this paper is that for any\u0000$epsilon>0$, three points in $S$ can be found on a horizontal line within\u0000distance $epsilon$ of each other. This gives a partial result toward a\u0000conjecture of Leli`evre. The set $S$ and group $Gamma$ are respectively the\u0000set of long cylinder holonomy vectors, and Veech group, of the \"golden L\"\u0000translation surface.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203833","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Our goal is to show that both the fast and slow versions of the triangle map (a type of multi-dimensional continued fraction algorithm) in dimension $n$ are ergodic, resolving a conjecture of Messaoudi, Noguiera and Schweiger. This particular type of higher dimensional multi-dimensional continued fraction algorithm has recently been linked to the study of partition numbers, with the result that the underlying dynamics has combinatorial implications.
{"title":"Ergodicity and Algebraticity of the Fast and Slow Triangle Maps","authors":"Thomas Garrity, Jacob Lehmann Duke","doi":"arxiv-2409.05822","DOIUrl":"https://doi.org/arxiv-2409.05822","url":null,"abstract":"Our goal is to show that both the fast and slow versions of the triangle map\u0000(a type of multi-dimensional continued fraction algorithm) in dimension $n$ are\u0000ergodic, resolving a conjecture of Messaoudi, Noguiera and Schweiger. This\u0000particular type of higher dimensional multi-dimensional continued fraction\u0000algorithm has recently been linked to the study of partition numbers, with the\u0000result that the underlying dynamics has combinatorial implications.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203780","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}