Pub Date : 2025-12-05DOI: 10.1007/s00039-025-00727-9
Ben Green
We give an asymptotic for the number of prime solutions to $Q(x_{1},dots , x_{8}) = N$Q(x1,…,x8)=N , subject to a mild non-degeneracy condition on the homogeneous quadratic form $Q$Q . The argument initially proceeds via the circle method, but this does not suffice by itself. To obtain a nontrivial bound on certain averages of exponential sums, we interpret these sums as matrix coefficients for the Weil representation of the symplectic group $operatorname {Sp}_{8}(mathbf{Z}/qmathbf{Z})$Sp8(Z/qZ) . Averages of such matrix coefficients are then bounded using an amplification argument and a convergence result for convolutions of measures, which reduces matters to understanding the action of certain 12-dimensional subgroups in the Weil representation. Sufficient understanding can be gained by using the basic represention theory of $operatorname {SL}_{2}(k)$SL2(k) , $k$k a finite field.
{"title":"Quadratic Forms in 8 Prime Variables","authors":"Ben Green","doi":"10.1007/s00039-025-00727-9","DOIUrl":"https://doi.org/10.1007/s00039-025-00727-9","url":null,"abstract":"We give an asymptotic for the number of prime solutions to <jats:inline-formula> <jats:alternatives> <jats:tex-math>$Q(x_{1},dots , x_{8}) = N$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>Q</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>8</mml:mn> </mml:msub> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>N</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> , subject to a mild non-degeneracy condition on the homogeneous quadratic form <jats:inline-formula> <jats:alternatives> <jats:tex-math>$Q$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>Q</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> . The argument initially proceeds via the circle method, but this does not suffice by itself. To obtain a nontrivial bound on certain averages of exponential sums, we interpret these sums as matrix coefficients for the Weil representation of the symplectic group <jats:inline-formula> <jats:alternatives> <jats:tex-math>$operatorname {Sp}_{8}(mathbf{Z}/qmathbf{Z})$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mo>Sp</mml:mo> <mml:mn>8</mml:mn> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>/</mml:mo> <mml:mi>q</mml:mi> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:math> </jats:alternatives> </jats:inline-formula> . Averages of such matrix coefficients are then bounded using an amplification argument and a convergence result for convolutions of measures, which reduces matters to understanding the action of certain 12-dimensional subgroups in the Weil representation. Sufficient understanding can be gained by using the basic represention theory of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$operatorname {SL}_{2}(k)$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mo>SL</mml:mo> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:math> </jats:alternatives> </jats:inline-formula> , <jats:inline-formula> <jats:alternatives> <jats:tex-math>$k$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>k</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> a finite field.","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"56 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145680180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-03DOI: 10.1007/s00039-025-00728-8
Huy Tuan Pham, Dmitrii Zakharov
A set of integers $A$A is non-averaging if there is no element $a$a in $A$A which can be written as an average of a subset of $A$A not containing $a$a . We show that the largest non-averaging subset of ${1, ldots , n}${1,…,n} has size $n^{1/4+o(1)}$n1/4+o(1) , thus solving the Erdős–Straus problem. We also determine the largest size of a non-averaging set in a $d$d -dimensional box for any fixed $d$d . Our main tool includes the structure theorem for the set of subset sums due to Conlon, Fox and the first author, together with a result about the structure of a point set in nearly convex position.
{"title":"Sharp Bound for the Erdős–Straus Non-averaging Set Problem","authors":"Huy Tuan Pham, Dmitrii Zakharov","doi":"10.1007/s00039-025-00728-8","DOIUrl":"https://doi.org/10.1007/s00039-025-00728-8","url":null,"abstract":"A set of integers <jats:inline-formula> <jats:alternatives> <jats:tex-math>$A$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>A</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> is <jats:italic>non-averaging</jats:italic> if there is no element <jats:inline-formula> <jats:alternatives> <jats:tex-math>$a$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>a</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> in <jats:inline-formula> <jats:alternatives> <jats:tex-math>$A$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>A</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> which can be written as an average of a subset of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$A$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>A</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> not containing <jats:inline-formula> <jats:alternatives> <jats:tex-math>$a$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>a</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> . We show that the largest non-averaging subset of <jats:inline-formula> <jats:alternatives> <jats:tex-math>${1, ldots , n}$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo>{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>}</mml:mo> </mml:math> </jats:alternatives> </jats:inline-formula> has size <jats:inline-formula> <jats:alternatives> <jats:tex-math>$n^{1/4+o(1)}$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn> <mml:mo>+</mml:mo> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> , thus solving the Erdős–Straus problem. We also determine the largest size of a non-averaging set in a <jats:inline-formula> <jats:alternatives> <jats:tex-math>$d$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>d</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> -dimensional box for any fixed <jats:inline-formula> <jats:alternatives> <jats:tex-math>$d$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>d</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> . Our main tool includes the structure theorem for the set of subset sums due to Conlon, Fox and the first author, together with a result about the structure of a point set in nearly convex position.","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"140 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145657528","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-28DOI: 10.1007/s00039-025-00726-w
Philip Easo
Let <jats:inline-formula> <jats:alternatives> <jats:tex-math>$(G_{n}) = left ((V_{n},E_{n})right )$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> be a sequence of finite connected vertex-transitive graphs with uniformly bounded vertex degrees such that <jats:inline-formula> <jats:alternatives> <jats:tex-math>$lvert V_{n} rvert to infty $</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>|</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>|</mml:mo> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> as <jats:inline-formula> <jats:alternatives> <jats:tex-math>$n to infty $</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> . We say that percolation on <jats:inline-formula> <jats:alternatives> <jats:tex-math>$G_{n}$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> has a <jats:italic>sharp</jats:italic> phase transition (as <jats:inline-formula> <jats:alternatives> <jats:tex-math>$n to infty $</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> ) if, as the percolation parameter crosses some critical point, the number of vertices contained in the largest percolation cluster jumps from logarithmic to linear order with high probability. We prove that percolation on <jats:inline-formula> <jats:alternatives> <jats:tex-math>$G_{n}$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> has a sharp phase transition unless, after passing to a subsequence, the rescaled graph-metric on <jats:inline-formula> <jats:alternatives> <jats:tex-math>$G_{n}$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> (rapidly) converges to the unit circle with respect to the Gromov-Hausdorff metric. We deduce that under the same hypothesis, the critical point for the emergence of a giant (i.e. linear-sized) cluster in <jats:inline-formula> <jats:alternatives> <jats:t
设$(G_{n}) = left ((V_{n},E_{n})right )$ （gn) = (vn, en)）是顶点度一致有界的有限连通顶点传递图序列，使得$lvert V_{n} rvert to infty $ | V n |→∞= $n to infty $ n→∞。如果当渗透参数越过某个临界点时，最大的渗透簇中包含的顶点数高概率地从对数阶跃到线性阶，我们说$G_{n}$ G n上的渗透具有急剧的相变（$n to infty $ n→∞）。我们证明了$G_{n}$ gn上的渗流有一个急剧的相变，除非在传递到子序列之后，$G_{n}$ gn上的重标图度量（迅速）收敛到相对于Gromov-Hausdorff度量的单位圆。我们推断，在相同的假设下，$G_{n}$ G n中出现巨大（即线性大小）集群的临界点与$(G_{n})$ （G n）的Benjamini-Schramm极限中出现无限集群的临界点重合，当该极限存在时。
{"title":"Sharpness and Locality for Percolation on Finite Transitive Graphs","authors":"Philip Easo","doi":"10.1007/s00039-025-00726-w","DOIUrl":"https://doi.org/10.1007/s00039-025-00726-w","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:tex-math>$(G_{n}) = left ((V_{n},E_{n})right )$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> be a sequence of finite connected vertex-transitive graphs with uniformly bounded vertex degrees such that <jats:inline-formula> <jats:alternatives> <jats:tex-math>$lvert V_{n} rvert to infty $</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo>|</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>|</mml:mo> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> as <jats:inline-formula> <jats:alternatives> <jats:tex-math>$n to infty $</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> . We say that percolation on <jats:inline-formula> <jats:alternatives> <jats:tex-math>$G_{n}$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> has a <jats:italic>sharp</jats:italic> phase transition (as <jats:inline-formula> <jats:alternatives> <jats:tex-math>$n to infty $</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> ) if, as the percolation parameter crosses some critical point, the number of vertices contained in the largest percolation cluster jumps from logarithmic to linear order with high probability. We prove that percolation on <jats:inline-formula> <jats:alternatives> <jats:tex-math>$G_{n}$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> has a sharp phase transition unless, after passing to a subsequence, the rescaled graph-metric on <jats:inline-formula> <jats:alternatives> <jats:tex-math>$G_{n}$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> (rapidly) converges to the unit circle with respect to the Gromov-Hausdorff metric. We deduce that under the same hypothesis, the critical point for the emergence of a giant (i.e. linear-sized) cluster in <jats:inline-formula> <jats:alternatives> <jats:t","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"1 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145610863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-19DOI: 10.1007/s00039-025-00724-y
Mingfeng Chen, Shengwen Gan, Shaoming Guo, Jonathan Hickman, Marina Iliopoulou, James Wright
We consider a class of Hörmander-type oscillatory integral operators in $mathbb{R}^{n}$Rn for $n geq 3$n≥3 odd with real analytic phase. We derive weak conditions on the phase which ensure $L^{p}$Lp bounds beyond the universal $p geq 2 cdot frac{n+1}{n-1}$p≥2⋅n+1n−1 range guaranteed by Stein’s oscillatory integral theorem. This expands and elucidates pioneering work of Bourgain from the early 1990s. We also consider a closely related class of variable coefficient Schrödinger propagator-type operators, and show that the corresponding theory differs significantly from that of the Hörmander-type operators. The main ingredient in the proof is a curved Kakeya/Nikodym maximal function estimate. This is established by combining the polynomial method with certain uniform sublevel set estimates for real analytic functions. The sublevel set estimates are the main novelty in the argument and can be interpreted as a form of quantification of linear independence in the $C^{omega }$Cω category.
考虑了在$mathbb{R}^{n}$ R n中，当$n geq 3$ n≥3奇时，一类具有实解析相的Hörmander-type振荡积分算子。我们在相位上导出了保证$L^{p}$ L p界超出Stein振荡积分定理所保证的$p geq 2 cdot frac{n+1}{n-1}$ p≥2⋅n + 1 n−1范围的弱条件。这是对20世纪90年代初布尔甘的开创性工作的扩展和阐释。我们还考虑了一类密切相关的变系数Schrödinger传播算子，并证明了相应的理论与Hörmander-type算子的理论有很大的不同。证明的主要成分是一个弯曲的Kakeya/Nikodym极大函数估计。这是将多项式方法与实解析函数的若干一致子水平集估计相结合建立起来的。子水平集估计是论证中的主要新颖之处，可以解释为$C^{omega }$ C ω类别中线性独立性的一种量化形式。
{"title":"Oscillatory Integral Operators and Variable Schrödinger Propagators: Beyond the Universal Estimates","authors":"Mingfeng Chen, Shengwen Gan, Shaoming Guo, Jonathan Hickman, Marina Iliopoulou, James Wright","doi":"10.1007/s00039-025-00724-y","DOIUrl":"https://doi.org/10.1007/s00039-025-00724-y","url":null,"abstract":"We consider a class of Hörmander-type oscillatory integral operators in <jats:inline-formula> <jats:alternatives> <jats:tex-math>$mathbb{R}^{n}$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula> <jats:alternatives> <jats:tex-math>$n geq 3$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:math> </jats:alternatives> </jats:inline-formula> odd with real analytic phase. We derive weak conditions on the phase which ensure <jats:inline-formula> <jats:alternatives> <jats:tex-math>$L^{p}$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> bounds beyond the universal <jats:inline-formula> <jats:alternatives> <jats:tex-math>$p geq 2 cdot frac{n+1}{n-1}$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>p</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> <mml:mo>⋅</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfrac> </mml:math> </jats:alternatives> </jats:inline-formula> range guaranteed by Stein’s oscillatory integral theorem. This expands and elucidates pioneering work of Bourgain from the early 1990s. We also consider a closely related class of variable coefficient Schrödinger propagator-type operators, and show that the corresponding theory differs significantly from that of the Hörmander-type operators. The main ingredient in the proof is a curved Kakeya/Nikodym maximal function estimate. This is established by combining the polynomial method with certain uniform sublevel set estimates for real analytic functions. The sublevel set estimates are the main novelty in the argument and can be interpreted as a form of quantification of linear independence in the <jats:inline-formula> <jats:alternatives> <jats:tex-math>$C^{omega }$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>ω</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> category.","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"19 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145545972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-31DOI: 10.1007/s00039-025-00723-z
Aditya Kumar, Balarka Sen
We construct infinitely many examples of macroscopically large manifolds of dimension $m geq 4$m≥4 equipped with circle bundles whose total spaces admit metrics of positive scalar curvature and have macroscopic dimension at most $lceil m/2 rceil + 1$⌈m/2⌉+1 . In particular, we answer a question of Gromov on the existence of circle bundles over enlargeable manifolds whose total spaces admit metrics of positive scalar curvature, in all dimensions. Our constructions are based on techniques from symplectic geometry.
{"title":"Circle Bundles with PSC over Large Manifolds","authors":"Aditya Kumar, Balarka Sen","doi":"10.1007/s00039-025-00723-z","DOIUrl":"https://doi.org/10.1007/s00039-025-00723-z","url":null,"abstract":"We construct infinitely many examples of macroscopically large manifolds of dimension <jats:inline-formula> <jats:alternatives> <jats:tex-math>$m geq 4$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>m</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:math> </jats:alternatives> </jats:inline-formula> equipped with circle bundles whose total spaces admit metrics of positive scalar curvature and have macroscopic dimension at most <jats:inline-formula> <jats:alternatives> <jats:tex-math>$lceil m/2 rceil + 1$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo>⌈</mml:mo> <mml:mi>m</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>⌉</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:math> </jats:alternatives> </jats:inline-formula> . In particular, we answer a question of Gromov on the existence of circle bundles over enlargeable manifolds whose total spaces admit metrics of positive scalar curvature, in all dimensions. Our constructions are based on techniques from symplectic geometry.","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"7 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145404392","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-09-01DOI: 10.1007/s00039-025-00718-w
Boaz Klartag, Joseph Lehec
We provide the final step in the resolution of Bourgain’s slicing problem in the affirmative. Thus we establish the following theorem: for any convex body K subseteq mathbb{R}^{n} of volume one, there exists a hyperplane H subseteq mathbb{R}^{n} such that
京公网安备 11010802042870号
京ICP备2023020795号-1
求助内容：
应助结果提醒方式：