We introduce a precise framework for transferring strategies from simpler to more complex games, and use it to construct strategies in certain finite and infinite combinations of games. In particular, we give a finitary characterization of finite hypergraphs X such that the first player can win the positional game on infinitely many copies of X. This resolves a conjecture of Leader.

We study ultrasymmetric spaces in the case in which the fundamental function belongs to a limiting class of quasiconcave functions. In the process, we study limiting cases of $J$ interpolation spaces and establish new $J$ – $K$ identities as well as a reiteration theorem for these limiting interpolation methods.

Arvanitakis [A simultaneous selection theorem. Preprint] recently established a theorem which is a common generalization of Michael’s convex selection theorem [Continuous selections I. Ann. of Math. (2) 63 (1956), 361–382] and Dugundji’s extension theorem [An extension of Tietze’s theorem, Pacific J. Math.1 (1951), 353–367]. In this note we provide a short proof of a more general version of Arvanitakis’s result.

A tensor-type integral formula for intrinsic volumes is used to define a further variant of directed projection functions and show that these determine a convex body uniquely. Averages of directed projection functions are then studied, and the connections between the resulting operators and previously considered spherical transforms discussed.

Answering two questions of Beresnevich and Velani, we develop zero-one laws in both simultaneous and multiplicative Diophantine approximation. Our proofs mainly rely on a versatile Cassels–Gallagher type theorem and the cross fibering principle of Beresnevich, Haynes and Velani.

Denote by ζk′(S), the derivative of the Dedekind zeta-function associated with the real quadratic field K. Then it is known that

where ζ(S) is the Riemann zeta-function, and L(s, x) is the Dirichlet L-series associated with the Legendre symbol X. Moreover, we have the functional equation

where D is the discriminant of K.

We prove a characterization of functions in B1/4(K)\C(K), where K is a compact metric space in terms of c0-spreading models, answering a Problem of R. Haydon, E. Odell and H. Rosenthal. Beginning with B1/4(K) we define a decreasing family (Vξ(K),║ · ║ξ)1≤ξ<ω1 of Banach spaces whose intersection is DBSC(K) and we prove an analogous stronger property for the functions in Vξ(K)\C(K). Defining the s-spreading model-index, we classify B1/4;(K) and we prove that s-SM[F]>ξ for every F∈ Vξ(K). Also we classify the separable Banach spaces by defining the c0-SM-index which measures the degree to which they have sequences with extending spreading models equivalent to the usual basis of c0. We give examples of Baire-1 functions and reflexive spaces with arbitrary large indices.

Let Λ′ be a lattice in three dimensional space. Let G be any point and denote by Λ the “displaced lattice”, consisting of all points X + G where X ε Λ′. Suppose that a sphere of radius 1 is centred at each point of Λ and a sphere of radius R−l, where 1 < R ≤ 2, is centred at each point of Λ′. If the lattice Λ′ and the point G are such that no two spheres of the system overlap we shall call Λ a mixed packing lattice, and shall denote its determinant by d(Λ) = d(Λ′). Let Δ be the lower bound of d(Λ) taken over all mixed packing lattices Λ. It can easily be proved that this is an attained lower bound, and any mixed packing lattice Λ, having d(Λ) = Δ, will be called a critical mixed packing lattice. We prove that

and shall describe, in Lemmas 1–3, the critical mixed packing lattices.

For d, n positive integers, let α(d, n) be the statement “For every abelian variety A bounded by d there exists a point P of order n”. Let l1, l2, l3, … be the sequence of prime numbers and let α be

The purpose of the paper is to introduce a novel “splitting” procedure which can be helpful in the derivation of explicit formulas for various Bellman functions. As an illustration, we study the action of the dyadic maximal operator on $L^{p}$. The associated Bellman function $\mathfrak{B}_{p}$, introduced by Nazarov and Treil, was found explicitly by Melas with the use of combinatorial properties of the maximal operator, and was later rediscovered by Slavin, Stokolos and Vasyunin with the use of the corresponding Monge–Ampère partial differential equation. Our new argument enables an alternative simple derivation of $\mathfrak{B}_{p}$.

Let K be a compact convex body in ℝn not contained in a hyperplane, and denote the norm whose unit ball is ½(K − K) by ║·║k. Given a translative packing of K, we are interested in how long a segment (with respect to ║·║K) can lie in the complement of the interiors of the translates. The main result of this note is to show the existence of a translative packing such that the length of the longest segments avoiding it is only exponential in the dimension n (see below). We start here with a lower bound, showing that this bound is close to optimal for balls.

For a positive integer $n$ let

$$\begin{eqnarray}\mathfrak{P}_{n}=\mathop{\prod }_{\substack{ p \\ s_{p}(n)\geqslant p}}p,\end{eqnarray}$$

$p$

$s_{p}(n)$

$p$

$n$

$n$

$\mathfrak{P}_{n}$

$\mathfrak{P}_{n}$

$\sqrt{n}$

$n^{20/37}$

$\sqrt{n}$

$\unicode[STIX]{x1D705}\sqrt{n}/\text{log}\,n$

$\unicode[STIX]{x1D705}$

$\unicode[STIX]{x1D705}=2$

$\mathfrak{P}_{n}$

$\mathfrak{P}_{n+1}$

$\mathfrak{P}_{n}$

$n$

$\mathfrak{P}_{n}>\mathfrak{P}_{n+1}$

A (countably) compact measure is one which is inner regular with respect to a (countably) compact class of sets. This note characterizes compact probability measures in terms of the representation of Boolean homomorphisms of their measure algebras, and shows that the same ideas can be used to give a direct proof of J. Pachl's theorem that any image measure of a countably compact measure is again countably compact.

L'objet de cet article est d'étudier un procédé de summation associé á certaines séries. Notant P(n) le plus grand facteur premier d'un entier générique n, nous rappellons les définitions de P-convergence et de P-régularité d'une série, introduites dans [7].

No systematic study seems to have been made of so natural a question as the analogue for matrices of quadratic residues. One generalization of x2 (x an integer) is X2 (X an integral matrix). Another is X′ X, where the prime means “transpose”. We study here the solvability for X of the congruence

where p is a prime, r ≥ 1; I (the identity matrix) and X are n-by-n; and a is an integer not divisible by p2.

A new and more geometric proof is obtained of Stanley's lower bounds on the face numbers of centrally symmetric simple polytopes.

Let

be a real quadratic form in n variables x1,…, xn and let ‖θ‖ denote the distance from θ to the nearest integer. Danicic [3] proved that if N > 1 and ξ > 0 then there exist integers x1,…, xn, not all zero, satisfying

where C depends on n and ξ only.

There is an error in the proof of Theorem 2 of my paper [1]. It appears on; age 91, lines 10 and 11: the application of the affine transformation T changes he measure on the Grassmannian G(d, d-i) which is not taken into account. As a result, in the statement of Theorem 2 the coefficient is not correct.

A reduction method due to R. Rado [7] yields an elegant proof of the Halls' theorems on transversals of a family of sets (see for example [4]). We use here this method to give simple new proofs of some basic theorems on common transversals of a pair of families of sets.

For a convex polytope P, let s(P) and σ(P) denote, respectively, the largest t such that P has a t-dimensional face (section) which is a simplex. Using Gale diagrams, lower bounds are obtained for these numbers and formulae are developed for the behaviour of s and σ over Cartesian and other products.