Given two sets of natural numbers A and B of natural density 1, we prove that their product set A · B : = { a b : a ∈ A , b ∈ B } also has natural density 1. On the other hand, for any ε > 0 , we show there are sets A of density > 1 − ε for which the product set A · A has density < ε . This answers two questions of Hegyvári, Hennecart and Pach.

We classify abelian subgroups of two‐dimensional Artin groups.

Let Ω ( n ) denote the number of prime factors of n . We show that for any bounded f : N → C one has 1 N ∑ n = 1 N f ( Ω ( n ) + 1 ) = 1 N ∑ n = 1 N f ( Ω ( n ) ) + o N → ∞ ( 1 ) . This yields a new elementary proof of the Prime Number Theorem.

We present a direct proof of asymptotic consensus in the non‐linear Hegselmann–Krause model with transmission‐type delay, where the communication weights depend on the particle distance in phase space. Our approach is based on an explicit estimate of the shrinkage of the group diameter on finite time intervals and avoids the usage of Lyapunov‐type functionals or results from non‐negative matrix theory

In a paper from 2001 (Journal of the LMS), Diestel and Leader offered a proof that a connected graph has a normal spanning tree if and only if it has no minor obtained canonically from either an ( ℵ 0 , ℵ 1 ) ‐regular bipartite graph or an order‐theoretic Aronszajn tree. In particular, this refuted an earlier conjecture of Halin's that only the first of these obstructions was needed to characterize

In the study of normal numbers using the base‐ b representation, b ⩾ 2 , we know that a number is normal if and only if it is simply normal in all the bases b n , n ⩾ 1 . We prove the analogue of this result for continued fraction normality. In particular, we show that two notions of continued fraction normality, one where overlapping occurrences of finite patterns are counted as distinct occurrences

Let G be a group definable in a weakly o‐minimal non‐valuational structure M . Then G / G 00 , equipped with the logic topology, is a compact Lie group, and if G has finitely satisfiable generics, then dim ( G / G 00 ) = dim ( G ) . Our main technical result is that G is a dense subgroup of a group definable in the canonical o‐minimal extension of M .

Given a Fano complete intersection defined by sections of a collection nef line bundles L 1 , … , L c on a Fano toric manifold Y , a construction of Givental/Hori–Vafa provides a mirror‐dual Landau–Ginzburg model. This construction depends on a choice of suitable nef partition; that is, a partition of the rays of the fan determined by Y . We show that Landau–Ginzburg models constructed from different

We prove a Cayley–Bacharach type theorem for points in projective space P n that lie on a complete intersection of n hypersurfaces. This is made possible by new bounds on the growth of the Hilbert function of almost complete intersections.

We describe a probabilistic model involving iterated Brownian motion for constructing a random chainable continuum. We show that this random continuum is indecomposable. We use our probabilistic model to define a Wiener‐type measure on the space of all chainable continua.

This work is motivated by the problem of finding locally compact group topologies for piecewise full groups (a.k.a. topological full groups). We determine that any piecewise full group that is locally compact in the compact‐open topology on the group of self‐homeomorphisms of the Cantor set must be uniformly discrete, in a precise sense that we introduce here. Uniformly discrete groups of self‐homeomorphisms

We construct a surface with a cylindrical end which has a finite number of Laplace eigenvalues embedded in its continuous spectrum. The surface is obtained by attaching a cylindrical end to a hyperbolic torus with a hole. To our knowledge, this is the first example of a manifold with a cylindrical end whose number of eigenvalues is known to be finite and nonzero. The construction can be varied to give

In the present work we have presented proof of Ball's integral inequality using symmetric B ‐spline functions. The proof is different from the before known.

We give two sufficient and necessary conditions for the Hochschild extension of a finite‐dimensional algebra by its dual bimodule and a Hochschild 2‐cocycle to be a symmetric algebra.

Let Ω be a C 4 ‐smooth bounded pseudoconvex domain in C 2 . We show that if the ∂ ¯ ‐Neumann operator N 1 is compact on L ( 0 , 1 ) 2 ( Ω ) , then the embedding operator J : D o m ( ∂ ¯ ) ∩ D o m ( ∂ ¯ ∗ ) → L ( 0 , 1 ) 2 ( Ω ) is L p ‐regular for all 2 ⩽ p < ∞ .

We prove that there exists at most one minimal diffeomorphism in a given homotopy class between any two closed Riemannian surfaces. This results was previously known only under the assumption that the Riemannian metrics have constant Gaussian curvature. Along the way, we prove the New Main Inequality which substantially strengthens the classical Reich–Strebel inequality for quasiconformal maps.

Let n ⩾ 6 be an integer. We prove that the number of number fields with Galois group A n and absolute discriminant at most X is asymptotically at least X 1 / 8 + O ( 1 / n ) . For n ⩾ 8 , this improves upon the previously best‐known lower bound of X ( 1 − 2 n ! ) / ( 4 n − 4 ) − ε , due to Pierce, Turnage‐Butterbaugh, and Wood.

A virtual link may be defined as an equivalence class of diagrams, or alternatively as a stable equivalence class of links in thickened surfaces. We prove that a minimal crossing virtual link diagram has minimal genus across representatives of the stable equivalence class. This is achieved by constructing a new parity theory for virtual links. As corollaries, we prove that the crossing, bridge, and

We show a Wolff–Denjoy type theorem in complete geodesic spaces in the spirit of Beardon's framework that unifies several results in this area. In particular, it applies to strictly convex bounded domains in R n or C n with respect to a large class of metrics including Hilbert's and Kobayashi's metrics. The results are generalized to 1‐Lipschitz compact mappings in infinite‐dimensional Banach spaces

In the article ‘Pseudo‐loop conditions’ by Pierre Gillibert, Julius Jonušas and Michael Pinsker [1], the authors would like to amend an incorrect grant number referenced in the Acknowledgement section. All authors have received funding from the Austrian Science Fund (FWF) through project No P27600. Michael Pinsker has received funding from the Czech Science Foundation grant No 18‐20123S.

The paper contains an alternative proof of the celebrated L p estimates for differentially subordinate martingales established by Burkholder and Wang in the eighties and nineties. The approach links the validity of the estimate to the existence of a lower solution to a novel boundary value problem.

Rickard proved that for certain self‐injective algebras, a stable equivalence induced from an exact functor is a stable equivalence of Morita type, in the sense of Broué. In this paper we study singular equivalences of finite‐dimensional algebras induced from tensor product functors. We prove that for certain Gorenstein algebras, a singular equivalence induced from tensoring with a suitable complex

For a prime p , we determine a Sylow p ‐subgroup D of a finite group G such that the principal p ‐block B of G has four irreducible ordinary characters. It has been determined already for the cases where the number is up to three by work by R. Brauer, J. Brandt, and V.A. Belonogov 30 years ago. Our proof relies on the classification of finite simple groups.

We prove some necessary conditions for a link to be either concordant to a quasi‐positive link, quasi‐positive, positive or the closure of a positive braid. The main applications of our results are a characterisation of positive links with unlinking numbers 1 and 2, and a combinatorial criterion to test if a positive link is the closure of a positive braid. Finally, we compile a table of all positive

We give a simple construction of new, complete, finite volume manifolds M of bounded, nonpositive curvature. These manifolds have ends that look like a mixture of locally symmetric ends of different ranks and their fundamental groups are not duality groups.

We show that the set of all possible constant diagonals of a bounded Hilbert space operator is always convex. This, in particular, answers an open question of Bourin (2003). Moreover, we show that the joint numerical range of a commuting operator tuple is, in general, not convex, which fills a gap in the literature. We also prove that the Asplund–Ptak numerical range (which is convex for pairs of operators)

Let σ = ( σ 1 , σ 2 , ⋯ , σ n ) ∈ S n − 1 and d σ denote the normalized Lebesgue measure on S n − 1 , n ⩾ 2 . For functions f 1 , f 2 , ⋯ , f n defined on R , consider the multilinear operator given by T ( f 1 , f 2 , ⋯ , f n ) ( x ) = ∫ S n − 1 ∏ j = 1 n f j ( x − σ j ) d σ , x ∈ R . In this paper, we obtain necessary and sufficient conditions on exponents p 1 , p 2 , ⋯ , p n and r for which the operator

We prove that the kernel of the evaluation morphism of global sections — namely the syzygy bundle — of a sufficiently ample line bundle on an abelian variety is stable. This settles a conjecture of Ein–Lazarsfeld–Mustopa, in the case of abelian varieties.

We provide a new and very short proof of the fact that a spherical functor between certain triangulated categories induces an auto‐equivalence.

Inspired by work of Lipshitz–Sarkar, we show that the module structure on link Floer homology detects split links. Using results of Ni, Alishahi–Lipshitz, and Lipshitz–Sarkar, we establish an analogous detection result for sutured Floer homology.

A classical result due to Eagleson states (in particular) that if appropriately normalized Birkhoff sums generated by a measurable function and an ergodic probability preserving transformation converge in distribution, then they also converge in distribution with respect to any probability measure which is absolutely continuous with respect to the invariant one. In this note, we prove several quantitative

We show that every bounded pseudo‐convex domain with Hölder boundary in C n is hyperconvex.

For a Lie algebra L , let S ( L ) denote the symmetric Poisson algebra and s ( L ) the truncated symmetric Poisson algebra of L . In characteristic p ≠ 2 , the conditions under which these Poisson algebras are solvable were established by Monteiro Alves and Petrogradsky in (J. Algebra 488 (2017) 244–281). The characterization in the harder case p = 2 was left as an open problem and a related conjecture

A matrix A is totally positive (or non‐negative) of order k , denoted T P k (or T N k ), if all minors of size ⩽ k are positive (or non‐negative). It is well known that such matrices are characterized by the variation diminishing property together with the sign non‐reversal property. We do away with the former, and show that A is T P k if and only if every submatrix formed from at most k consecutive

Let K be a complete, discretely valued field with finite residue field and G K its absolute Galois group. The subject of this note is the study of the set of positive integers d for which there exists an absolutely irreducible ℓ ‐adic representation of G K of dimension d with rational traces on inertia. Our main result is that non‐Sophie Germain primes are not in this set when the residue characteristic

We show that the Fraïssé limit of a category of unital separable C ∗ ‐algebras which is sufficiently closed under tensor products of its objects and morphisms is strongly self‐absorbing, given that it has approximately inner half‐flip. We use this connection between Fraïssé limits and strongly self‐absorbing C ∗ ‐algebras to give a rather elementary proof for the well‐known fact that the Jiang–Su algebra

For all 1 ⩽ m ⩽ n − 1 , we investigate the interaction of locally finite measures in R n with the family of m ‐dimensional Lipschitz graphs. For instance, we characterize Radon measures μ , which are carried by Lipschitz graphs in the sense that there exist graphs Γ 1 , Γ 2 , ⋯ such that μ ( R n ∖ ⋃ 1 ∞ Γ i ) = 0 , using only countably many evaluations of the measure. This problem in geometric measure

Hepworth, Willerton, Leinster and Shulman introduced the magnitude homology groups for enriched categories, in particular, for metric spaces. The purpose of this paper is to describe the magnitude homology group of a metric space in terms of order complexes of posets.

Let D be the unit disk. Kutzschebauch and Studer (Bull. Lond. Math. Soc. 51 (2019) 995–1004) recently proved that, for each continuous map A : D ¯ → SL ( 2 , C ) , which is holomorphic in D , there exist continuous maps E , F : D ¯ → sl ( 2 , C ) , which are holomorphic in D , such that A = e E e F . Also they asked if this extends to arbitrary compact bordered Riemann surfaces. We prove that this

In a celebrated paper, Borwein, Erdélyi, Ferguson and Lockhart constructed cosine polynomials of the form f A ( x ) = ∑ a ∈ A cos ( a x ) , with A ⊆ N , | A | = n and as few as n 5 / 6 + o ( 1 ) zeros in [ 0 , 2 π ] , thereby disproving an old conjecture of Littlewood. Here we give a sharp analysis of their constructions and, as a result, prove that there exist examples with as few as C ( n log n )

In the 1970s, Joan Birman and Hugh Hilden wrote several papers on the problem of relating the mapping class group of a surface to that of a covering space. Their results provide a bridge between the theories of mapping class groups and braid groups. We survey the work of Birman and Hilden, give an overview of the subsequent developments, and discuss open questions and new directions.

We use Dowlin's spectral sequence from Khovanov homology to knot Floer homology to prove that reduced Khovanov homology with rational coefficients detects the figure‐eight knot.

We prove new upper bounds on the number of representations of rational numbers m n as a sum of four unit fractions, giving five different regions, depending on the size of m in terms of n . In particular, we improve the most relevant cases, when m is small, and when m is close to n . The improvements stem from not only studying complete parametrizations of the set of solutions, but simplifying this

We give necessary and sufficient conditions for a 4‐manifold to be a branched covering of C P 2 , S 2 × S 2 , S 2 × ∼ S 2 or S 3 × S 1 , which are expressed in terms of the Betti numbers and the signature of the 4‐manifold. Moreover, we extend these results to include branched coverings of connected sums of the above manifolds. This leads to some new examples of closed simply connected quasiregularly

We verify numerically, in a rigorous way using interval arithmetic, that the Riemann hypothesis is true up to height 3 · 10 12 . That is, all zeroes β + i γ of the Riemann zeta‐function with 0 < γ ⩽ 3 · 10 12 have β = 1 / 2 . Moreover, all of these zeroes are simple.

We give upper bounds for several highness properties in computability randomness theory. First, we prove that a certain discrete covering property (which requires to almost cover all sets in a certain class) does not imply the ability to compute a 1‐random real, answering a question of Greenberg, Miller and Nies. This also implies that an infinite set of incompressible strings does not necessarily

We find the sharp range for boundedness of the discrete bilinear spherical maximal function for dimensions d ⩾ 5 . That is, we show that this operator is bounded on l p ( Z d ) × l q ( Z d ) → l r ( Z d ) for 1 / p + 1 / q ⩾ 1 / r and r > d / ( d − 2 ) and we show this range is sharp. Our approach mirrors that used by Jeong and Lee in the continuous setting. For dimensions d = 3 , 4 , our previous

In this paper we compute the intersection number of two double ramification (DR) cycles (with different ramification profiles) and the top Chern class of the Hodge bundle on the moduli space of stable curves of any genus. These quadratic DR integrals are the main ingredients for the computation of the DR hierarchy associated to the infinite‐dimensional partial cohomological field theory given by exp

We show that the imaginary multiplicative chaos exp ( i β Γ ) determines the gradient of the underlying field Γ for all log‐correlated Gaussian fields with covariance of the form − log | x − y | + g ( x , y ) with mild regularity conditions on g , for all d ⩾ 2 and for all β ∈ ( 0 , d ) . In particular, we show that the 2D continuum zero boundary Gaussian free field is measurable with respect to its

In this paper, we study conjugacy classes for pivotal fusion categories. In particular, we prove a Burnside type formula for the structure constants concerning the product of two conjugacy class sums of such a fusion category. For a braided weakly integral fusion category C , we show that these structure constants multiplied by dim ( C ) are non‐negative integers, extending some results obtained by

For an irreducible character χ of a finite group G , the codegree of χ is defined by | G : ker χ | / χ ( 1 ) . In this note, we show that if a finite solvable group G has an element of order m , then G admits an irreducible character of codegree divisible by m .

Coxeter matroids generalize matroids just as flag varieties of Lie groups generalize Grassmannians. Valuations of Coxeter matroids are functions that behave well with respect to subdivisions of a Coxeter matroid into smaller ones. We compute the universal valuative invariant of Coxeter matroids. A key ingredient is the family of Coxeter Schubert matroids, which correspond to the Bruhat cells of flag

For every positive integer n , we construct a Hasse diagram with n vertices and independence number O ( n 3 / 4 ) . Such graphs have chromatic number Ω ( n 1 / 4 ) , which significantly improves the previously best‐known constructions of Hasse diagrams having chromatic number Θ ( log n ) . In addition, if we also require girth of at least k ⩾ 5 , we construct such Hasse diagrams with independence number

We construct new finite semigroups in β N , in particular, an m ‐element null semigroup for any m ⩾ 2 and a 7‐element semigroup generated by two idempotents. We also point out a connection of finite semigroups in β N with Ramsey theory. In particular, we deduce that for every m ⩾ 2 , there exists a partition { A 1 , … , A m } of N with the following property: for any finite partitions B i of A i ,

We prove a contractive Hardy–Littlewood type inequality for functions from H p ( T ) , 0 < p ⩽ 2 which is sharp in the first two Taylor coefficients and asymptotically at infinity.

We investigate a variant of the Beurling–Ahlfors extension of quasisymmetric homeomorphisms of the real line that is given by the convolution of the heat kernel, and prove that the complex dilatation of such a quasiconformal extension of a strongly symmetric homeomorphism (that is, its derivative is an A ∞ ‐weight whose logarithm is in VMO) induces a vanishing Carleson measure on the upper half‐plane

We show the existence of non‐Einstein cones which are critical for all quadratic curvature functionals.

We determine the diffraction intensity of the k ‐free integers near the origin.

We continue our study of ‘no‐dimension’ analogues of basic theorems in combinatorial and convex geometry in Banach spaces. We generalize some results of the paper (Adiprasito, Bárány and Mustafa, ‘Theorems of Carathéodory, Helly, and Tverberg without dimension’, Proceedings of the Thirtieth Annual ACM‐SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, San Diego,

Let X be a compact Kähler manifold of dimension n . For 1 ⩽ j ⩽ m , let T j be a closed positive (1,1)‐current on X whose cohomology class is Kähler. Let T be a closed positive current of bi‐degree ( p , p ) with m + p ⩽ n . We prove that if T 1 , … , T m are of T ‐relative full mass intersection, then the T ‐relative non‐pluripolar product of T 1 , … , T m is equal to the Dinh–Sibony product of T