样式: 排序: IF: - GO 导出 标记为已读
-
The sup-norm problem beyond the newform Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2024-03-12 EDGAR ASSING
In this paper we take up the classical sup-norm problem for automorphic forms and view it from a new angle. Given a twist minimal automorphic representation $\pi$ we consider a special small $\mathrm{GL}_2(\mathbb{Z}_p)$ -type V in $\pi$ and prove global sup-norm bounds for an average over an orthonormal basis of V. We achieve a non-trivial saving when the dimension of V grows.
-
A bound of the number of weighted blow-ups to compute the minimal log discrepancy for smooth 3-folds Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2024-03-08 SHIHOKO ISHII
We study a pair consisting of a smooth 3-fold defined over an algebraically closed field and a “general” ${\Bbb R}$ -ideal. We show that the minimal log discrepancy (“mld” for short) of every such a pair is computed by a prime divisor obtained by at most two weighted blow-ups. This bound is regarded as a weighted blow-up version of Mustaţă–Nakamura’s conjecture. We also show that if the mld of such
-
How to solve a binary cubic equation in integers Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2024-03-07 DAVID MASSER
Given any polynomial in two variables of degree at most three with rational integer coefficients, we obtain a new search bound to decide effectively if it has a zero with rational integer coefficients. On the way we encounter a natural problem of estimating singular points. We solve it using elementary invariant theory but an optimal solution would seem to be far from easy even using the full power
-
Log Calabi–Yau surfaces and Jeffrey–Kirwan residues Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2024-03-04 RICCARDO ONTANI, JACOPO STOPPA
We prove an equality, predicted in the physical literature, between the Jeffrey–Kirwan residues of certain explicit meromorphic forms attached to a quiver without loops or oriented cycles and its Donaldson–Thomas type invariants. In the special case of complete bipartite quivers we also show independently, using scattering diagrams and theta functions, that the same Jeffrey–Kirwan residues are determined
-
The uniform distribution modulo one of certain subsequences of ordinates of zeros of the zeta function Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2024-03-01 FATMA ÇİÇEK, STEVEN M. GONEK
On the assumption of the Riemann hypothesis and a spacing hypothesis for the nontrivial zeros $1/2+i\gamma$ of the Riemann zeta function, we show that the sequence \begin{equation*}\Gamma_{[a, b]} =\Bigg\{ \gamma : \gamma>0 \quad \mbox{and} \quad \frac{ \log\big(| \zeta^{(m_{\gamma })} (\frac12+ i{\gamma }) | / (\!\log{{\gamma }} )^{m_{\gamma }}\big)}{\sqrt{\frac12\log\log {\gamma }}} \in [a, b] \Bigg\}
-
Zeros, chaotic ratios and the computational complexity of approximating the independence polynomial Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-11-24 DAVID DE BOER, PJOTR BUYS, LORENZO GUERINI, HAN PETERS, GUUS REGTS
The independence polynomial originates in statistical physics as the partition function of the hard-core model. The location of the complex zeros of the polynomial is related to phase transitions, and plays an important role in the design of efficient algorithms to approximately compute evaluations of the polynomial. In this paper we directly relate the location of the complex zeros of the independence
-
Three Schur functors related to pre-Lie algebras Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-10-16 VLADIMIR DOTSENKO, OISÍN FLYNN-CONNOLLY
We give explicit combinatorial descriptions of three Schur functors arising in the theory of pre-Lie algebras. The first of them leads to a functorial description of the underlying vector space of the universal enveloping pre-Lie algebra of a given Lie algebra, strengthening the Poincaré-Birkhoff-Witt (PBW) theorem of Segal. The two other Schur functors provide functorial descriptions of the underlying
-
On subdirect products of type FPn of limit groups over Droms RAAGs Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-10-11 DESSISLAVA H. KOCHLOUKOVA, JONE LOPEZ DE GAMIZ ZEARRA
We generalise some known results for limit groups over free groups and residually free groups to limit groups over Droms RAAGs and residually Droms RAAGs, respectively. We show that limit groups over Droms RAAGs are free-by-(torsion-free nilpotent). We prove that if S is a full subdirect product of type $FP_s(\mathbb{Q})$ of limit groups over Droms RAAGs with trivial center, then the projection of
-
Abelian tropical covers Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-10-10 YOAV LEN, MARTIN ULIRSCH, DMITRY ZAKHAROV
Let $\mathfrak{A}$ be a finite abelian group. In this paper, we classify harmonic $\mathfrak{A}$-covers of a tropical curve $\Gamma$ (which allow dilation along edges and at vertices) in terms of the cohomology group of a suitably defined sheaf on $\Gamma$. We give a realisability criterion for harmonic $\mathfrak{A}$-covers by patching local monodromy data in an extended homology group on $\Gamma$
-
On Bohr compactifications and profinite completions of group extensions Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-10-09 BACHIR BEKKA
Let $G= N\rtimes H$ be a locally compact group which is a semi-direct product of a closed normal subgroup N and a closed subgroup H. The Bohr compactification ${\rm Bohr}(G)$ and the profinite completion ${\rm Prof}(G)$ of G are, respectively, isomorphic to semi-direct products $Q_1 \rtimes {\rm Bohr}(H)$ and $Q_2 \rtimes {\rm Prof}(H)$ for appropriate quotients $Q_1$ of ${\rm Bohr}(N)$ and $Q_2$ of
-
Nonvarying, affine and extremal geometry of strata of differentials Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-10-06 DAWEI CHEN
We prove that the nonvarying strata of abelian and quadratic differentials in low genus have trivial tautological rings and are affine varieties. We also prove that strata of k-differentials of infinite area are affine varieties for all k. Vanishing of homology in degree higher than the complex dimension follows as a consequence for these affine strata. Moreover we prove that the stratification of
-
On the topology of the transversal slice of a quasi-homogeneous map germ Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-10-06 O. N. SILVA
We consider a corank 1, finitely determined, quasi-homogeneous map germ f from $(\mathbb{C}^2,0)$ to $(\mathbb{C}^3,0)$. We describe the embedded topological type of a generic hyperplane section of $f(\mathbb{C}^2)$, denoted by $\gamma_f$, in terms of the weights and degrees of f. As a consequence, a necessary condition for a corank 1 finitely determined map germ $g\,{:}\,(\mathbb{C}^2,0)\rightarrow
-
A problem of Erdős–Graham–Granville–Selfridge on integral points on hyperelliptic curves Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-10-05 HUNG M. BUI, KYLE PRATT, ALEXANDRU ZAHARESCU
Erdős, Graham and Selfridge considered, for each positive integer n, the least value of $t_n$ so that the integers $n+1, n+2, \dots, n+t_n $ contain a subset the product of whose members with n is a square. An open problem posed by Granville concerns the size of $t_n$, under the assumption of the ABC conjecture. We establish some results on the distribution of $t_n$, and in the process solve Granville’s
-
Spirals of Riemann’s Zeta-Function — Curvature, Denseness and Universality Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-10-05 ATHANASIOS SOURMELIDIS, JÖRN STEUDING
This paper deals with applications of Voronin’s universality theorem for the Riemann zeta-function $\zeta$. Among other results we prove that every plane smooth curve appears up to a small error in the curve generated by the values $\zeta(\sigma+it)$ for real t where $\sigma\in(1/2,1)$ is fixed. In this sense, the values of the zeta-function on any such vertical line provides an atlas for plane curves
-
Structure of fine Selmer groups over -extensions Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-10-04 MENG FAI LIM
This paper is concerned with the study of the fine Selmer group of an abelian variety over a $\mathbb{Z}_{p}$-extension which is not necessarily cyclotomic. It has been conjectured that these fine Selmer groups are always torsion over $\mathbb{Z}_{p}[[ \Gamma ]]$, where $\Gamma$ is the Galois group of the $\mathbb{Z}_{p}$-extension in question. In this paper, we shall provide several strong evidences
-
Multiple recurrence and popular differences for polynomial patterns in rings of integers Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-10-02 ETHAN ACKELSBERG, VITALY BERGELSON
We demonstrate that the phenomenon of popular differences (aka the phenomenon of large intersections) holds for natural families of polynomial patterns in rings of integers of number fields. If K is a number field with ring of integers $\mathcal{O}_K$ and $E \subseteq \mathcal{O}_K$ has positive upper Banach density $d^*(E) = \delta > 0$, we show, inter alia: (1) if $p(x) \in K[x]$ is an intersective
-
Some torsion-free solvable groups with few subquotients Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-10-02 ADRIEN LE BOUDEC, NICOLÁS MATTE BON
We construct finitely generated torsion-free solvable groups G that have infinite rank, but such that all finitely generated torsion-free metabelian subquotients of G are virtually abelian. In particular all finitely generated metabelian subgroups of G are virtually abelian. The existence of such groups shows that there is no “torsion-free version” of P. Kropholler’s theorem, which characterises solvable
-
Categories of graphs for operadic structures Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-09-28 PHILIP HACKNEY
We recall several categories of graphs which are useful for describing homotopy-coherent versions of generalised operads (e.g. cyclic operads, modular operads, properads, and so on), and give new, uniform definitions for their morphisms. This allows for straightforward comparisons, and we use this to show that certain free-forgetful adjunctions between categories of generalised operads can be realised
-
Lower bounds on the maximal number of rational points on curves over finite fields Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-09-28 JONAS BERGSTRÖM, EVERETT W. HOWE, ELISA LORENZO GARCÍA, CHRISTOPHE RITZENTHALER
For a given genus $g \geq 1$, we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus g over $\mathbb{F}_q$. As a consequence of Katz–Sarnak theory, we first get for any given $g>0$, any $\varepsilon>0$ and all q large enough, the existence of a curve of genus g over $\mathbb{F}_q$ with at least $1+q+ (2g-\varepsilon) \sqrt{q}$ rational
-
Non-Orientable Lagrangian Fillings of Legendrian Knots Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-09-27 LINYI CHEN, GRANT CRIDER-PHILLIPS, BRAEDEN REINOSO, JOSHUA SABLOFF, LEYU YAO
We investigate when a Legendrian knot in the standard contact ${{\mathbb{R}}}^3$ has a non-orientable exact Lagrangian filling. We prove analogs of several results in the orientable setting, develop new combinatorial obstructions to fillability, and determine when several families of knots have such fillings. In particular, we completely determine when an alternating knot (and more generally a plus-adequate
-
Galois groups and prime divisors in random quadratic sequences Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-09-07 JOHN R. DOYLE, VIVIAN OLSIEWSKI HEALEY, WADE HINDES, RAFE JONES
Given a set $S=\{x^2+c_1,\dots,x^2+c_s\}$ defined over a field and an infinite sequence $\gamma$ of elements of S, one can associate an arboreal representation to $\gamma$, generalising the case of iterating a single polynomial. We study the probability that a random sequence $\gamma$ produces a “large-image” representation, meaning that infinitely many subquotients in the natural filtration are maximal
-
Dirichlet law for factorisation of integers, polynomials and permutations Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-09-06 SUN–KAI LEUNG
Let $k \geqslant 2$ be an integer. We prove that factorisation of integers into k parts follows the Dirichlet distribution $\mathrm{Dir}\left({1}/{k},\ldots,{1}/{k}\right)$ by multidimensional contour integration, thereby generalising the Deshouillers–Dress–Tenenbaum (DDT) arcsine law on divisors where $k=2$. The same holds for factorisation of polynomials or permutations. Dirichlet distribution with
-
Equidistribution of exponential sums indexed by a subgroup of fixed cardinality Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-08-24 THÉO UNTRAU
We consider families of exponential sums indexed by a subgroup of invertible classes modulo some prime power q. For fixed d, we restrict to moduli q so that there is a unique subgroup of invertible classes modulo q of order d. We study distribution properties of these families of sums as q grows and we establish equidistribution results in some regions of the complex plane which are described as the
-
On the values taken by slice torus invariants Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-08-23 PETER FELLER, LUKAS LEWARK, ANDREW LOBB
We study the space of slice torus invariants. In particular we characterise the set of values that slice torus invariants may take on a given knot in terms of the stable smooth slice genus. Our study reveals that the resolution of the local Thom conjecture implies the existence of slice torus invariants without having to appeal to any explicit construction from a knot homology theory.
-
Weil Sums over Small Subgroups Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-08-15 ALINA OSTAFE, IGOR E. SHPARLINSKI, JOSÉ FELIPE VOLOCH
We obtain new bounds on short Weil sums over small multiplicative subgroups of prime finite fields which remain nontrivial in the range the classical Weil bound is already trivial. The method we use is a blend of techniques coming from algebraic geometry and additive combinatorics.
-
Projection theorems for linear-fractional families of projections Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-08-08 ANNINA ISELI, ANTON LUKYANENKO
Marstrand’s theorem states that applying a generic rotation to a planar set A before projecting it orthogonally to the x-axis almost surely gives an image with the maximal possible dimension $\min(1, \dim A)$. We first prove, using the transversality theory of Peres–Schlag locally, that the same result holds when applying a generic complex linear-fractional transformation in $PSL(2,\mathbb{C})$ or
-
Complex multiplication and Noether–Lefschetz loci of the twistor space of a K3 surface Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-07-27 FRANCESCO VIGANÒ
For an algebraic K3 surface with complex multiplication (CM), algebraic fibres of the associated twistor space away from the equator are again of CM type. In this paper, we show that algebraic fibres corresponding to points at the same altitude of the twistor base ${S^2} \simeq \mathbb{P}_\mathbb{C}^1$ share the same CM endomorphism field. Moreover, we determine all the admissible Picard numbers of
-
Projections of the minimal nilpotent orbit in a simple Lie algebra and secant varieties Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-07-13 DMITRI PANYUSHEV
Let G be a simple algebraic group with ${\mathfrak g}={\textrm{Lie }} G$ and ${\mathcal O}_{\textsf{min}}\subset{\mathfrak g}$ the minimal nilpotent orbit. For a ${\mathbb Z}_2$-grading ${\mathfrak g}={\mathfrak g}_0\oplus{\mathfrak g}_1$, let $G_0$ be a connected subgroup of G with ${\textrm{Lie }} G_0={\mathfrak g}_0$. We study the $G_0$-equivariant projections $\varphi\,:\,\overline{{\mathcal O
-
Prime divisors and the number of conjugacy classes of finite groups Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-07-10 THOMAS MICHAEL KELLER, ALEXANDER MORETÓ
We prove that there exists a universal constant D such that if p is a prime divisor of the index of the Fitting subgroup of a finite group G, then the number of conjugacy classes of G is at least $Dp/\log_2p$. We conjecture that we can take $D=1$ and prove that for solvable groups, we can take $D=1/3$.
-
Multifractal analysis of sums of random pulses Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-06-22 GUILLAUME SAËS, STÉPHANE SEURET
In this paper, we investigate the regularity properties and determine the almost sure multifractal spectrum of a class of random functions constructed as sums of pulses with random dilations and translations. In addition, the continuity moduli of the sample paths of these stochastic processes are investigated.
-
Necessary condition for the L2 boundedness of the Riesz transform on Heisenberg groups Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-05-23 DAMIAN DĄBROWSKI, MICHELE VILLA
Let $\mu$ be a Radon measure on the nth Heisenberg group ${\mathbb{H}}^n$. In this note we prove that if the $(2n+1)$-dimensional (Heisenberg) Riesz transform on ${\mathbb{H}}^n$ is $L^2(\mu)$-bounded, and if $\mu(F)=0$ for all Borel sets with ${\text{dim}}_H(F)\leq 2$, then $\mu$ must have $(2n+1)$-polynomial growth. This is the Heisenberg counterpart of a result of Guy David from [Dav91].
-
Sign changes of fourier coefficients of holomorphic cusp forms at norm form arguments Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-05-23 ALEXANDER P. MANGEREL
Let f be a non-CM Hecke eigencusp form of level 1 and fixed weight, and let $\{\lambda_f(n)\}_n$ be its sequence of normalised Fourier coefficients. We show that if $K/ \mathbb{Q}$ is any number field, and $\mathcal{N}_K$ denotes the collection of integers representable as norms of integral ideals of K, then a positive proportion of the positive integers $n \in \mathcal{N}_K$ yield a sign change for
-
On the integral Hodge conjecture for varieties with trivial Chow group Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-05-17 HUMBERTO A. DIAZ
We obtain examples of smooth projective varieties over ${\mathbb C}$ that violate the integral Hodge conjecture and for which the total Chow group is of finite rank. Moreover, we show that there exist such examples defined over number fields.
-
Coniveau filtrations and Milnor operation Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-05-08 NOBUAKI YAGITA
Let BG be the classifying space of an algebraic group G over the field ${\mathbb C}$ of complex numbers. There are smooth projective approximations X of $BG\times {\mathbb P}^{\infty}$, by Ekedahl. We compute a new stable birational invariant of X defined by the difference of two coniveau filtrations of X, by Benoist and Ottem. Hence we give many examples such that two coniveau filtrations are different
-
Zariski dense orbits for regular self-maps of split semiabelian varieties in positive characteristic Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-05-02 DRAGOS GHIOCA, SINA SALEH
We prove the Zariski dense orbit conjecture in positive characteristic for regular self-maps of split semiabelian varieties.
-
Positive lower density for prime divisors of generic linear recurrences Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-04-24 OLLI JÄRVINIEMI
Let $d \ge 3$ be an integer and let $P \in \mathbb{Z}[x]$ be a polynomial of degree d whose Galois group is $S_d$. Let $(a_n)$ be a non-degenerate linearly recursive sequence of integers which has P as its characteristic polynomial. We prove, under the generalised Riemann hypothesis, that the lower density of the set of primes which divide at least one non-zero element of the sequence $(a_n)$ is positive
-
A characterisation of atomicity Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-04-24 SALVATORE TRINGALI
In a 1968 issue of the Proceedings, P. M. Cohn famously claimed that a commutative domain is atomic if and only if it satisfies the ascending chain condition on principal ideals (ACCP). Some years later, a counterexample was however provided by A. Grams, who showed that every commutative domain with the ACCP is atomic, but not vice versa. This has led to the problem of finding a sensible (ideal-theoretic)
-
Relatively hyperbolic groups with strongly shortcut parabolics are strongly shortcut Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-04-17 NIMA HODA, SURAJ KRISHNA M S
We show that a group that is hyperbolic relative to strongly shortcut groups is itself strongly shortcut, thus obtaining new examples of strongly shortcut groups. The proof relies on a result of independent interest: we show that every relatively hyperbolic group acts properly and cocompactly on a graph in which the parabolic subgroups act properly and cocompactly on convex subgraphs.
-
The Picard group of vertex affinoids in the first Drinfeld covering Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-04-13 JAMES TAYLOR
Let F be a finite extension of ${\mathbb Q}_p$. Let $\Omega$ be the Drinfeld upper half plane, and $\Sigma^1$ the first Drinfeld covering of $\Omega$. We study the affinoid open subset $\Sigma^1_v$ of $\Sigma^1$ above a vertex of the Bruhat–Tits tree for $\text{GL}_2(F)$. Our main result is that $\text{Pic}\!\left(\Sigma^1_v\right)[p] = 0$, which we establish by showing that $\text{Pic}({\mathbf Y})[p]
-
Random amenable C*-algebras Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-04-12 BHISHAN JACELON
What is the probability that a random UHF algebra is of infinite type? What is the probability that a random simple AI algebra has at most k extremal traces? What is the expected value of the radius of comparison of a random Villadsen-type AH algebra? What is the probability that such an algebra is $\mathcal{Z}$-stable? What is the probability that a random Cuntz–Krieger algebra is purely infinite
-
Equations of mirrors to log Calabi–Yau pairs via the heart of canonical wall structures Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-04-11 HÜLYA ARGÜZ
Gross and Siebert developed a program for constructing in arbitrary dimension a mirror family to a log Calabi–Yau pair (X, D), consisting of a smooth projective variety X with a normal-crossing anti-canonical divisor D in X. In this paper, we provide an algorithm to practically compute explicit equations of the mirror family in the case when X is obtained as a blow-up of a toric variety along hypersurfaces
-
Intermediate-scale statistics for real-valued lacunary sequences Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-04-11 NADAV YESHA
We study intermediate-scale statistics for the fractional parts of the sequence $\left(\alpha a_{n}\right)_{n=1}^{\infty}$, where $\left(a_{n}\right)_{n=1}^{\infty}$ is a positive, real-valued lacunary sequence, and $\alpha\in\mathbb{R}$. In particular, we consider the number of elements $S_{N}\!\left(L,\alpha\right)$ in a random interval of length $L/N$, where $L=O\!\left(N^{1-\epsilon}\right)$, and
-
The Log Product Formula in Quantum K-theory Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-04-11 YOU–CHENG CHOU, LEO HERR, YUAN–PIN LEE
We prove a formula expressing the K-theoretic log Gromov-Witten invariants of a product of log smooth varieties $V \times W$ in terms of the invariants of V and W. The proof requires introducing log virtual fundamental classes in K-theory and verifying their various functorial properties. We introduce a log version of K-theory and prove the formula there as well.
-
Stable finiteness does not imply linear soficity Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-04-11 BE’ERI GREENFELD
We prove that there exist finitely generated, stably finite algebras which are not linear sofic. This was left open by Arzhantseva and Păunescu in 2017.
-
Paucity problems and some relatives of Vinogradov’s mean value theorem Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-04-05 TREVOR D. WOOLEY
When $k\geqslant 4$ and $0\leqslant d\leqslant (k-2)/4$, we consider the system of Diophantine equations \begin{align*}x_1^j+\ldots +x_k^j=y_1^j+\ldots +y_k^j\quad (1\leqslant j\leqslant k,\, j\ne k-d).\end{align*}We show that in this cousin of a Vinogradov system, there is a paucity of non-diagonal positive integral solutions. Our quantitative estimates are particularly sharp when $d=o\!\left(k^{1/4}\right)$
-
Finite point configurations in products of thick Cantor sets and a robust nonlinear Newhouse Gap Lemma Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-03-13 ALEX MCDONALD, KRYSTAL TAYLOR
In this paper we prove that the set $\{|x^1-x^2|,\dots,|x^k-x^{k+1}|\,{:}\,x^i\in E\}$ has non-empty interior in $\mathbb{R}^k$ when $E\subset \mathbb{R}^2$ is a Cartesian product of thick Cantor sets $K_1,K_2\subset\mathbb{R}$. We also prove more general results where the distance map $|x-y|$ is replaced by a function $\phi(x,y)$ satisfying mild assumptions on its partial derivatives. In the process
-
Co-t-structures, cotilting and cotorsion pairs Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-03-10 DAVID PAUKSZTELLO, ALEXANDRA ZVONAREVA
Let $\textsf{T}$ be a triangulated category with shift functor $\Sigma \colon \textsf{T} \to \textsf{T}$ . Suppose $(\textsf{A},\textsf{B})$ is a co-t-structure with coheart $\textsf{S} = \Sigma \textsf{A} \cap \textsf{B}$ and extended coheart $\textsf{C} = \Sigma^2 \textsf{A} \cap \textsf{B} = \textsf{S}* \Sigma \textsf{S}$ , which is an extriangulated category. We show that there is a bijection between
-
Motivic zeta functions of hyperplane arrangements Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-03-07 MAX KUTLER, JEREMY USATINE
For each central essential hyperplane arrangement $\mathcal{A}$ over an algebraically closed field, let $Z_\mathcal{A}^{\hat\mu}(T)$ denote the Denef–Loeser motivic zeta function of $\mathcal{A}$ . We prove a formula expressing $Z_\mathcal{A}^{\hat\mu}(T)$ in terms of the Milnor fibers of related hyperplane arrangements. This formula shows that, in a precise sense, the degree to which $Z_{\mathcal{A}}^{\hat\mu}(T)$
-
Boolean algebras, Morita invariance and the algebraic K-theory of Lawvere theories Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-02-27 ANNA MARIE BOHMANN, MARKUS SZYMIK
The algebraic K-theory of Lawvere theories is a conceptual device to elucidate the stable homology of the symmetry groups of algebraic structures such as the permutation groups and the automorphism groups of free groups. In this paper, we fully address the question of how Morita equivalence classes of Lawvere theories interact with algebraic K-theory. On the one hand, we show that the higher algebraic
-
On some hyperelliptic Hurwitz–Hodge integrals Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-02-23 DANILO LEWAŃSKI
We address Hodge integrals over the hyperelliptic locus. Recently Afandi computed, via localisation techniques, such one-descendant integrals and showed that they are Stirling numbers. We give another proof of the same statement by a very short argument, exploiting Chern classes of spin structures and relations arising from Topological Recursion in the sense of Eynard and Orantin. These techniques
-
Null-homotopic knots have Property R Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-02-20 YI NI
We prove that if K is a nontrivial null-homotopic knot in a closed oriented 3–manfiold Y such that $Y-K$ does not have an $S^1\times S^2$ summand, then the zero surgery on K does not have an $S^1\times S^2$ summand. This generalises a result of Hom and Lidman, who proved the case when Y is an irreducible rational homology sphere.
-
Intersection theorems for finite general linear groups Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-02-16 ALENA ERNST, KAI–UWE SCHMIDT
A subset Y of the general linear group $\text{GL}(n,q)$ is called t-intersecting if $\text{rk}(x-y)\le n-t$ for all $x,y\in Y$ , or equivalently x and y agree pointwise on a t-dimensional subspace of $\mathbb{F}_q^n$ for all $x,y\in Y$ . We show that, if n is sufficiently large compared to t, the size of every such t-intersecting set is at most that of the stabiliser of a basis of a t-dimensional subspace
-
Divisors computing minimal log discrepancies on lc surfaces Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-02-14 JIHAO LIU, LINGYAO XIE
Let $(X\ni x,B)$ be an lc surface germ. If $X\ni x$ is klt, we show that there exists a divisor computing the minimal log discrepancy of $(X\ni x,B)$ that is a Kollár component of $X\ni x$ . If $B\not=0$ or $X\ni x$ is not Du Val, we show that any divisor computing the minimal log discrepancy of $(X\ni x,B)$ is a potential lc place of $X\ni x$ . This extends a result of Blum and Kawakita who independently
-
Relative Richardson varieties Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-02-14 MELODY CHAN, NATHAN PFLUEGER
A Richardson variety in a flag variety is an intersection of two Schubert varieties defined by transverse flags. We define and study relative Richardson varieties, which are defined over a base scheme with a vector bundle and two flags. To do so, we generalise transversality of flags to a relative notion, versality, that allows the flags to be non-transverse over some fibers. Relative Richardson varieties
-
Equivariant Heegaard genus of reducible 3-manifolds Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-02-13 SCOTT A. TAYLOR
The equivariant Heegaard genus of a 3-manifold W with the action of a finite group G of diffeomorphisms is the smallest genus of an equivariant Heegaard splitting for W. Although a Heegaard splitting of a reducible manifold is reducible and although if W is reducible, there is an equivariant essential sphere, we show that equivariant Heegaard genus may be super-additive, additive, or sub-additive under
-
Simultaneous p-adic Diophantine approximation Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2023-01-31 V. BERESNEVICH, J. LEVESLEY, B. WARD
The aim of this paper is to develop the theory of weighted Diophantine approximation of rational numbers to p-adic numbers. Firstly, we establish complete analogues of Khintchine’s theorem, the Duffin–Schaeffer theorem and the Jarník–Besicovitch theorem for ‘weighted’ simultaneous Diophantine approximation in the p-adic case. Secondly, we obtain a lower bound for the Hausdorff dimension of weighted
-
A zero density estimate and fractional imaginary parts of zeros for L-functions Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2022-12-28 OLIVIA BECKWITH, DI LIU, JESSE THORNER, ALEXANDRU ZAHARESCU
We prove an analogue of Selberg’s zero density estimate for $\zeta(s)$ that holds for any $\textrm{GL}_2$ L-function. We use this estimate to study the distribution of the vector of fractional parts of $\gamma\boldsymbol{\alpha}$ , where $\boldsymbol{\alpha}\in\mathbb{R}^n$ is fixed and $\gamma$ varies over the imaginary parts of the nontrivial zeros of a $\textrm{GL}_2$ L-function.
-
Lower bounds on Hilbert–Kunz multiplicities and maximal F-signatures Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2022-12-21 JACK JEFFRIES, YUSUKE NAKAJIMA, ILYA SMIRNOV, KEI–ICHI WATANABE, KEN–ICHI YOSHIDA
Hilbert–Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild assumptions. A natural question is for what singular rings these invariants are closest to one. For Hilbert–Kunz multiplicity this question was first considered
-
Finite quasisimple groups acting on rationally connected threefolds Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2022-12-06 JÉRÉMY BLANC, IVAN CHELTSOV, ALEXANDER DUNCAN, YURI PROKHOROV
We show that the only finite quasi-simple non-abelian groups that can faithfully act on rationally connected threefolds are the following groups: ${\mathfrak{A}}_5$ , ${\text{PSL}}_2(\textbf{F}_7)$ , ${\mathfrak{A}}_6$ , ${\text{SL}}_2(\textbf{F}_8)$ , ${\mathfrak{A}}_7$ , ${\text{PSp}}_4(\textbf{F}_3)$ , ${\text{SL}}_2(\textbf{F}_{7})$ , $2.{\mathfrak{A}}_5$ , $2.{\mathfrak{A}}_6$ , $3.{\mathfrak{A}}_6$
-
Pseudo-holomorphic dynamics in the restricted three-body problem Math. Proc. Camb. Philos. Soc. (IF 0.8) Pub Date : 2022-12-05 AGUSTIN MORENO
In this paper, we identify the five dimensional analogue of the finite energy foliations introduced by Hofer–Wysocki–Zehnder for the study of three dimensional Reeb flows, and show that these exist for the spatial circular restricted three-body problem (SCR3BP) whenever the planar dynamics is convex. We introduce the notion of a fiberwise-recurrent point, which may be thought of as a symplectic version