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  • Pair correlation of sequences $(\lbrace a_n \alpha \rbrace)_{n \in {\mathbb N}}$ with maximal additive energy
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-09-06
    GERHARD LARCHER; WOLFGANG STOCKINGER

    We show for sequences $\left(a_{n}\right)_{n \in \mathbb N}$ of distinct positive integers with maximal order of additive energy, that the sequence $\left(\left\{a_{n} \alpha\right\}\right)_{n \in \mathbb N}$ does not have Poissonian pair correlations for any α. This result essentially sharpens a result obtained by J. Bourgain on this topic.

    更新日期:2020-02-21
  • Local rings with quasi-decomposable maximal ideal
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-09-26
    SAEED NASSEH; RYO TAKAHASHI

    Let (R, 𝔪) be a commutative noetherian local ring. In this paper, we prove that if 𝔪 is decomposable, then for any finitely generated R-module M of infinite projective dimension 𝔪 is a direct summand of (a direct sum of) syzygies of M. Applying this result to the case where 𝔪 is quasi-decomposable, we obtain several classifications of subcategories, including a complete classification of the thick subcategories of the singularity category of R.

    更新日期:2020-02-21
  • Additive correlation and the inverse problem for the large sieve
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-07-09
    BRANDON HANSON

    Let A ⊆ [1, N] be a set of integers with |A| ≫ $\sqrt N$ . We show that if A avoids about p/2 residue classes modulo p for each prime p, then A must correlate additively with the squares S = {n2 : 1 ≤ n ≤ $\sqrt N$ }, in the sense that we have the additive energy estimate $$ E(A,S)\gg N\log N. $$ This is, in a sense, optimal.

    更新日期:2020-02-11
  • On the Saxl graph of a permutation group
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-08-08
    TIMOTHY C. BURNESS; MICHAEL GIUDICI

    Let G be a permutation group on a set Ω. A subset of Ω is a base for G if its pointwise stabiliser in G is trivial. In this paper we introduce and study an associated graph Σ(G), which we call the Saxl graph of G. The vertices of Σ(G) are the points of Ω, and two vertices are adjacent if they form a base for G. This graph encodes some interesting properties of the permutation group. We investigate the connectivity of Σ(G) for a finite transitive group G, as well as its diameter, Hamiltonicity, clique and independence numbers, and we present several open problems. For instance, we conjecture that if G is a primitive group with a base of size 2, then the diameter of Σ(G) is at most 2. Using a probabilistic approach, we establish the conjecture for some families of almost simple groups. For example, the conjecture holds when G = Sn or An (with n > 12) and the point stabiliser of G is a primitive subgroup. In contrast, we can construct imprimitive groups whose Saxl graph is disconnected with arbitrarily many connected components, or connected with arbitrarily large diameter.

    更新日期:2020-02-11
  • Remarks on Wolff's inequality for hypersurfaces
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-09-06
    SHAOMING GUO; CHANGKEUN OH

    We run an iteration argument due to Pramanik and Seeger, to provide a proof of sharp decoupling inequalities for conical surfaces and for k-cones. These are extensions of results of Łaba and Pramanik to sharp exponents.

    更新日期:2020-02-11
  • On orbifold constructions associated with the Leech lattice vertex operator algebra
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-09-05
    CHING HUNG LAM; HIROKI SHIMAKURA

    In this paper, we study orbifold constructions associated with the Leech lattice vertex operator algebra. As an application, we prove that the structure of a strongly regular holomorphic vertex operator algebra of central charge 24 is uniquely determined by its weight one Lie algebra if the Lie algebra has the type A3,43A1,2, A4,52, D4,12A2,6, A6,7, A7,4A1,13, D5,8A1,2 or D6,5A1,12 by using the reverse orbifold construction. Our result also provides alternative constructions of these vertex operator algebras (except for the case A6,7) from the Leech lattice vertex operator algebra.

    更新日期:2020-02-11
  • Classification of the real discrete specialisations of the Burau representation of B3
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-10-09
    NANCY SCHERICH

    This classification is found by analyzing the action of a normal subgroup of B3 as hyperbolic isometries. This paper gives an example of an unfaithful specialisation of the Burau representation on B4 that is faithful when restricted to B3, as well as examples of unfaithful specialisations of B3.

    更新日期:2020-02-11
  • The frame of smashing tensor-ideals
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-10-19
    PAUL BALMER; HENNING KRAUSE; GREG STEVENSON

    We prove that every flat tensor-idempotent in the module category Mod- of a tensor-triangulated category comes from a unique smashing ideal in . We deduce that the lattice of smashing ideals forms a frame.

    更新日期:2020-02-11
  • The structure of triple homomorphisms onto prime algebras
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-10-23
    CHENG–KAI LIU

    Triple homomorphisms on C*-algebras and JB*-triples have been studied in the literature. From the viewpoint of associative algebras, we characterise the structure of triple homomorphisms from an arbitrary ⋆-algebra onto a prime *-algebra. As an application, we prove that every triple homomorphism from a Banach ⋆-algebra onto a prime semisimple idempotent Banach *-algebra is continuous. The analogous results for prime C*-algebras and standard operator *-algebras on Hilbert spaces are also described.

    更新日期:2020-02-11
  • Rational discrete first degree cohomology for totally disconnected locally compact groups
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-10-12
    ILARIA CASTELLANO

    It is well known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group G can be detected on the cohomology group H1(G,R[G]), where R is either a finite field, the ring of integers or the field of rational numbers. It will be shown (cf. Theorem A*) that for a compactly generated totally disconnected locally compact group G the same information about the number of ends of G in the sense of H. Abels can be provided by dH1(G, Bi(G)), where Bi(G) is the rational discrete standard bimodule of G, and dH•(G, _) denotes rational discrete cohomology as introduced in [6].

    更新日期:2020-02-11
  • Sequences, modular forms and cellular integrals
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-10-11
    DERMOT McCARTHY; ROBERT OSBURN; ARMIN STRAUB

    It is well-known that the Apéry sequences which arise in the irrationality proofs for ζ(2) and ζ(3) satisfy many intriguing arithmetic properties and are related to the pth Fourier coefficients of modular forms. In this paper, we prove that the connection to modular forms persists for sequences associated to Brown's cellular integrals and state a general conjecture concerning supercongruences.

    更新日期:2020-02-11
  • A group version of stable regularity
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-10-24
    G. CONANT; A. PILLAY; C. TERRY

    We prove that, given ε > 0 and k ≥ 1, there is an integer n such that the following holds. Suppose G is a finite group and A ⊆ G is k-stable. Then there is a normal subgroup H ≤ G of index at most n, and a set Y ⊆ G, which is a union of cosets of H, such that |A △ Y| ≤ε|H|. It follows that, for any coset C of H, either |C ∩ A|≤ ε|H| or |C \ A| ≤ ε |H|. This qualitatively generalises recent work of Terry and Wolf on vector spaces over $\mathbb{F}_p$ .

    更新日期:2020-02-11
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  • On the iteration of quasimeromorphic mappings
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-07-06
    LUKE WARREN

    The Fatou–Julia theory for rational functions has been extended both to transcendental meromorphic functions and more recently to several different types of quasiregular mappings in higher dimensions. We extend the iterative theory to quasimeromorphic mappings with an essential singularity at infinity and at least one pole, constructing the Julia set for these maps. We show that this Julia set shares many properties with those for transcendental meromorphic functions and for quasiregular mappings of punctured space.

    更新日期:2020-01-04
  • Krull's principal ideal theorem in non-Noetherian settings
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-08-08
    BRUCE OLBERDING

    Let P be a finitely generated ideal of a commutative ring R. Krull's principal ideal theorem states that if R is Noetherian and P is minimal over a principal ideal of R, then P has height at most one. Straightforward examples show that this assertion fails if R is not Noetherian. We consider what can be asserted in the non-Noetherian case in place of Krull's theorem.

    更新日期:2020-01-04
  • Geodesic X-ray tomography for piecewise constant functions on nontrapping manifolds
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-09-12
    JOONAS ILMAVIRTA; JERE LEHTONEN; MIKKO SALO

    We show that on a two-dimensional compact nontrapping manifold with strictly convex boundary, a piecewise constant function is determined by its integrals over geodesics. In higher dimensions, we obtain a similar result if the manifold satisfies a foliation condition. These theorems are based on iterating a local uniqueness result. Our proofs are elementary.

    更新日期:2020-01-04
  • Median and injective metric spaces
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-07-27
    BRIAN H. BOWDITCH

    We describe a construction which associates to any median metric space a pseudometric satisfying the binary intersection property for closed balls. Under certain conditions, this implies that the resulting space is, in fact, an injective metric space, bilipschitz equivalent to the original metric. In the course of doing this, we derive a few other facts about median metrics, and the geometry of CAT(0) cube complexes. One motivation for the study of such metrics is that they arise as asymptotic cones of certain naturally occurring spaces.

    更新日期:2020-01-04
  • Small sets containing any pattern
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-07-31
    URSULA MOLTER; ALEXIA YAVICOLI

    Given any dimension function h, we construct a perfect set E ⊆ ${\mathbb{R}}$ of zero h-Hausdorff measure, that contains any finite polynomial pattern.

    更新日期:2020-01-04
  • Regulator constants of integral representations of finite groups
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-09-05
    ALEX TORZEWSKI

    Let G be a finite group and p be a prime. We investigate isomorphism invariants of $\mathbb{Z}_p$ [G]-lattices whose extension of scalars to $\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally introduced by Dokchitser–Dokchitser in the context of elliptic curves. Regulator constants canonically yield a pairing between the space of Brauer relations for G and the subspace of the representation ring for which regulator constants are defined. For all G, we show that this pairing is never identically zero. For formal reasons, this pairing will, in general, have non-trivial kernel. But, if G has cyclic Sylow p-subgroups and we restrict to considering permutation lattices, then we show that the pairing is non-degenerate modulo the formal kernel. Using this we can show that, for certain groups, including dihedral groups of order 2p for p odd, the isomorphism class of any $\mathbb{Z}_p$ [G]-lattice whose extension of scalars to $\mathbb{Q}_p$ is self-dual, is determined by its regulator constants, its extension of scalars to $\mathbb{Q}_p$ , and a cohomological invariant of Yakovlev.

    更新日期:2020-01-04
  • Interior of sums of planar sets and curves
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-09-05
    KÁROLY SIMON; KRYSTAL TAYLOR

    Recently, considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the interior (A + Γ)°, when Γ is a piecewise ${\mathcal C}^2$ curve and A ⊂ ℝ2. To begin, we give an example of a very large (full-measure, dense, Gδ) set A such that (A + S1)° = ∅, where S1 denotes the unit circle. This suggests that merely the size of A does not guarantee that (A + S1)° ≠ ∅. If, however, we assume that A is a kind of generalised product of two reasonably large sets, then (A + Γ)° ≠ ∅ whenever Γ has non-vanishing curvature. As a byproduct of our method, we prove that the pinned distance set of C := Cγ × Cγ, γ ⩾ 1/3, pinned at any point of C has non-empty interior, where Cγ (see (1.1)) is the middle 1 − 2γ Cantor set (including the usual middle-third Cantor set, C1/3). Our proof for the middle-third Cantor set requires a separate method. We also prove that C + S1 has non-empty interior.

    更新日期:2020-01-04
  • The distribution of consecutive prime biases and sums of sawtooth random variables
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-08-02
    ROBERT J. LEMKE OLIVER; KANNAN SOUNDARARAJAN

    In recent work, we considered the frequencies of patterns of consecutive primes (mod q) and numerically found biases toward certain patterns and against others. We made a conjecture explaining these biases, the dominant factor in which permits an easy description but fails to distinguish many patterns that have seemingly very different frequencies. There was a secondary factor in our conjecture accounting for this additional variation, but it was given only by a complicated expression whose distribution was not easily understood. Here, we study this term, which proves to be connected to both the Fourier transform of classical Dedekind sums and the error term in the asymptotic formula for the sum of φ(n).

    更新日期:2020-01-04
  • Supercongruences for truncated hypergeometric series and p-adic gamma function
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-08-10
    RUPAM BARMAN; NEELAM SAIKIA

    We prove three more general supercongruences between truncated hypergeometric series and p-adic gamma function from which some known supercongruences follow. A supercongruence conjectured by Rodriguez--Villegas and proved by E. Mortenson using the theory of finite field hypergeometric series follows from one of our more general supercongruences. We also prove a supercongruence for 7F6 truncated hypergeometric series which is similar to a supercongruence proved by L. Long and R. Ramakrishna.

    更新日期:2020-01-04
  • On the weak forms of the 2-part of Birch and Swinnerton-Dyer conjecture
    Math. Proc. Camb. Philos. Soc. (IF 0.737) Pub Date : 2018-09-05
    SHUAI ZHAI

    In this paper, we investigate the weak forms of the 2-part of the conjecture of Birch and Swinnerton-Dyer, and prove a lower bound for the 2-adic valuation of the algebraic part of the central value of the complex L-series for the family of quadratic twists of all optimal elliptic curves over ${\mathbb Q}$ .

    更新日期:2020-01-04
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