Bulletin of the London Mathematical Society最新文献

Assuming the generalized Lindelöf hypothesis (GLH), a weak version of the generalized Ramanujan conjecture and a Rankin–Selberg type partial sum estimate, we establish the normality of the sum of coefficients of a general $�L$-function in short intervals of appropriate length. The novelty lies in the degree aspect under GLH. In particular, this generalizes the result of Hughes and Rudnick on lattice point counts in thin annuli.

We prove a codimension reduction and congruence theorem for compact $�n$-dimensional submanifolds of $�S^{�n�+�p�}$ that admit a mean convex isometric embedding into $�S_{+}^{�n�+�1�}$ using a Reilly-type formula for space forms.

We introduce adic tropicalizations for subschemes of toric varieties as limits of Gubler models associated to polyhedral covers of the ordinary tropicalization. Our main result shows that Huber's adic analytification of a subscheme of a toric variety is naturally isomorphic to the inverse limit of its adic tropicalizations, in the category of locally topologically ringed spaces. The key new technical idea underlying this theorem is cofinality of Gubler models, which we prove for projective schemes and also for more general compact analytic domains in closed subschemes of toric varieties. In addition, we introduce a $�G$-topology and structure sheaf on ordinary tropicalizations, and show that Berkovich analytifications are limits of ordinary tropicalizations in the category of topologically ringed topoi.

We study the singular series associated to a cubic form with integer coefficients. If the number of variables is at least 10, we prove the absolute convergence (and hence positivity) under the assumption of Davenport's Geometric Condition, improving on a result of Heath-Brown. For the case of nine variables, we give a conditional treatment. We also provide a new short and elementary proof of Davenport's Shrinking Lemma that has been a crucial tool in previous literature on this and related problems.

This note considers the finite linear independence of coherent systems associated to discrete subgroups. We show by simple arguments that such coherent systems of amenable groups are linearly independent whenever the associated twisted group ring does not contain any nontrivial zero divisors. We verify the latter for discrete subgroups in nilpotent Lie groups. For the particular case of time-frequency translates of Euclidean space, our approach provides a simple and self-contained proof of the Heil–Ramanathan–Topiwala conjecture for subsets of arbitrary discrete subgroups.

In this paper, we study the universal $�m$-gonal forms. More precisely, we study the growth of the size of the finite set $��\{�1�,�2�,�\text{…}�,�{\gamma }_m�\}�$ ($�{\gamma }_m$ asymptotically increases as $�m$ increases) which characterize the universality of $�m$-gonal forms.

We show that the resonant frequencies of a system of coupled resonators in a truncated periodic lattice converge to the essential spectrum of the corresponding infinite lattice. We use the capacitance matrix as a model for fully coupled subwavelength resonators with long-range interactions in three spatial dimensions. For one-, two- or three-dimensional lattices embedded in three-dimensional space, we show that the (discrete) density of states (DOS) for the finite system converges in distribution to the (continuous) DOS of the infinite system. We achieve this by proving a weak convergence of the finite capacitance matrix to corresponding (translationally invariant) Toeplitz matrix of the infinite structure. With this characterisation at hand, we use the truncated Floquet transform to introduce a notion of spectral band structure for finite materials. This principle is also applicable to structures that are not translationally invariant and have interfaces. We demonstrate this by considering examples of perturbed systems with defect modes, such as an analogue of the well-known interface Su–Schrieffer–Heeger model.

We prove that the Lelek fan admits a completely scrambled weakly mixing homeomorphism. This is then used to show that for every $�n$ there is a continuum of topological dimension $�n$ admitting a weakly mixing completely scrambled homeomorphism. This provides a final answer to a question from 2001.

This paper provides answers to several open problems about equational theories of idempotent semifields. In particular, it is proved that (i) no equational theory of a non-trivial class of idempotent semifields has a finite basis; (ii) there are continuum-many equational theories of classes of idempotent semifields; and (iii) the equational theory of the class of idempotent semifields is co-NP-complete. This last result is also used to determine the complexity of deciding the existence of a right order on a free group or free monoid satisfying finitely many given inequalities.