Israel Journal of Mathematics最新文献

The first-order part of Ramsey’s theorem for pairs with an arbitrary number of colors is known to be precisely BΣ⁰₃. We compare this to the known division of Ramsey’s theorem for pairs into the weaker principles, EM (the Erdős–Moser principle) and ADS (the ascending-descending sequence principle): we show that the additional strength beyond IΣ⁰₂ is entirely due to the arbitrary color analog of ADS.

Specifically, we show that ADS for an arbitrary number of colors implies BΣ⁰₃ while EM for an arbitrary number of colors is Π¹₁-conservative over IΣ⁰₂ and it does not imply IΣ⁰₂.

A Markov operator P on a probability space (S, Σ μ) with μ invariant, is called hyperbounded if for some 1 ≤ p≤ q ≤ ∞ it maps (continuously) L^p into L^q.

We deduce from a recent result of Glück that a hyperbounded P is quasi-compact, hence uniformly ergodic, in all L^r(S, μ), 1 < r < ∞. We prove, using a method similar to Foguel’s, that a hyperbounded Markov operator has periodic behavior similar to that of Harris recurrent operators, and for the ergodic case obtain conditions for aperiodicity.

Given a probability ν on the unit circle, we prove that if the convolution operator P_νf:= ν ⋇ f is hyperbounded, then ν is atomless. We show that there is ν absolutely continuous such that P_ν is not hyperbounded, and there is ν with all powers singular such that P_ν is hyperbounded. As an application, we prove that if P_ν is hyperbounded, then for any sequence (n_k) of distinct positive integers with bounded gaps, (n_kx) is uniformly distributed mod 1 for ν almost every x (even when ν is singular).

A linear system L over (mathbb{F}_{q}) is common if the number of monochromatic solutions to L = 0 in any two-colouring of (mathbb{F}_{q}^{n}) is asymptotically at least the expected number of monochromatic solutions in a random two-colouring of (mathbb{F}_{q}^{n}). Motivated by existing results for specific systems (such as Schur triples and arithmetic progressions), as well as extensive research on common and Sidorenko graphs, Saad and Wolf recently initiated the systematic study of common systems of linear equations.

Building upon earlier work of Cameron, Cilleruelo and Serra, as well as Saad and Wolf, common linear equations have recently been fully characterised by Fox, Pham and Zhao, who asked about common systems of equations. In this paper we move towards a classification of common systems of two or more linear equations. In particular we prove that any system containing an arithmetic progression of length four is uncommon, resolving a question of Saad and Wolf. This follows from a more general result which allows us to deduce the uncommonness of a general system from certain properties of one- or two-equation subsystems.

The f-invariant is a notion of entropy for probability-measure-preserving actions of free groups. We show it is invariant under bounded orbit-equivalence.

We study the potential of Borel asymptotic dimension, a tool introduced recently in [2], to help produce Borel edge colorings of Schreier graphs generated by Borel group actions. We find that it allows us to recover the classical bound of Vizing in certain cases, and also use it to exactly determine the Borel edge chromatic number for free actions of abelian groups.

Let Σ_g,p be an oriented surface of genus g with p punctures. We denote by (cal{M}_{g,p}) and (cal{M}_{g,p}^{pm}) the mapping class group and the extended mapping class group of Σ_g,p, respectively. In this paper, we show that (cal{M}_{g,p}) and (cal{M}_{g,p}^{pm}) are generated by two elements for g ≥ 3 and p ≥ 0.

To an orthogonal or unitary involution on a central simple algebra of degree 4, or to a symplectic involution on a central simple algebra of degree 8, we associate a Pfister form that characterises the decomposability of the algebra with involution. In this way we obtain a unified approach to known decomposability criteria for several cases, and a new result for symplectic involutions on degree-8 algebras in characteristic 2.

We show that an integral non-degenerate fusion category ({cal C}) is group-theoretical if the Frobenius–Perron dimensions of its simple objects are either 1 or powers of a prime p.

The Blaschke–Santaló inequality is a classical inequality in convex geometry concerning the volume of a convex body and that of its dual. In this work we investigate an analogue of this inequality in the context of a billiard dynamical system: we replace the volume with the length of the shortest closed billiard trajectory. We define a quantity called the “billiard product” of a convex body K, which is analogous to the volume product studied in the Blaschke–Santaló inequality. In the planar case, we derive an explicit expression for the billiard product in terms of the diameter of the body. We also investigate upper bounds for this quantity in the class of polygons with a fixed number of vertices.

We introduce and study a class of multivariate rational functions associated with hyperplane arrangements, called flag Hilbert–Poincaré series. These series are intimately connected with Igusa local zeta functions of products of linear polynomials, and their motivic and topological relatives. Our main results include a self-reciprocity result for central arrangements defined over fields of characteristic zero. We also prove combinatorial formulae for a specialization of the flag Hilbert–Poincaré series for irreducible Coxeter arrangements of types A, B, and D in terms of total partitions of the respective types. We show that a different specialization of the flag Hilbert–Poincaré series, which we call the coarse flag Hilbert–Poincaré series, exhibits intriguing nonnegativity features and—in the case of Coxeter arrangements—connections with Eulerian polynomials. For numerous classes and examples of hyperplane arrangements, we determine their (coarse) flag Hilbert–Poincaré series. Some computations were aided by a SageMath package we developed.