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 solvability of convolution equations for functions with discrete support in $\mathbf{R}^{n}$, a special case being functions with support in the integer points. The more general case is of interest for several grids in Euclidean space, like the body-centred and face-centred tessellations of 3-space, as well as for the non-periodic grids that appear in the study of quasicrystals. The theorem of existence of fundamental solutions by de Boor et al is generalized to general discrete supports, using only elementary methods. We also study the asymptotic growth of sequences and arrays using the Fenchel transformation.

In the above paper [1] there is an error in the statements of Proposition 1 and Theorem 1 which we shall now correct. The principal results, Theorems 2 and 3, are unaffected.

We develop Weyl differencing and Hua-type lemmata for a class of multidimensional exponential sums. We then apply our estimates to bound the number of variables required to establish an asymptotic formula for the number of solutions of a system of diophantine equations arising from the study of linear spaces on hypersurfaces. For small values of the degree and dimension, our results are superior to those stemming from the author’s earlier work on Vinogradov’s mean value theorem.

A sequence {an} of integers is said to be primitive if ai × aj whenever i ≠ j. Let f be a polynomial with integer coefficients and A a sequence of positive integers. We discuss further a problem considered in [1] in which I. Anderson, W. W. Stothers and the author investigated primitive sequences of the form f(A) = {f(x), x ∈ A}. (Of course, we can assume f(x)→ ∞ as x → ∞.) We shall prove the following theorem in which A(n), as usual, denotes the number of memhers of A that are. less than or equal to n.

This paper is motivated by Davenport’s problem and the subsequent work regarding badly approximable points in submanifolds of a Euclidean space. We study the problem in the area of twisted Diophantine approximation and present two different approaches. The first approach shows that, under a certain restriction, any countable intersection of the sets of weighted badly approximable points on any non-degenerate ${\mathcal{C}}^{1}$ submanifold of $\mathbb{R}^{n}$ has full dimension. In the second approach, we introduce the property of isotropically winning and show that the sets of weighted badly approximable points are isotropically winning under the same restriction as above.

Let K be an algebraic number field, [K: Q] = κ є N; only the case κ > 1 is of interest in this paper. Let f be any non-zero ideal in ZK, the ring of integers of K, and let b be any ray-class (modx f) of K. In this paper we answer a question of P. Erdös (private communication) about the “maximum-growth-rate” of the functions

and

the sum here taken over all ray-classes (modx f), while N(a) is the absolute norm of a. Let

and

where, as usual, for x є R, log+ x - log max {1, x}. We prove

THEOREM 1. For every f, b we have

1. If w(mod 2ω1, 2ω2) is an elliptic parameter for points of a normal elliptic curve C = 1Cn[n − 1], then it is well known that the sets of n points in which C is met by primes have a constant parameter sum k (mod 2ωl, 2ω2), and we may express this for convenience by saying that k is the prime parameter sum for the parametrisation of C by w. If we take the origin of w (the point for which w ≡ 0) to be one of the points of hyperosculation of C, then k ≡ 0, and we may say that w is a normal parameter for C. In the same way, if Γ is the Grassmannian image curve of the generators of a normal elliptic scroll 1R2n[n − 1], then a normal parametrisation of Γ defines a normal parameter w for the generators of 1R2n, such that n of the generators have parameter sum zero if and only if they belong to a linear line-complex not containing all the generators of 1R2n; or, in particular, if they all meet a space [n – 3] that is not met by every generator of the scroll. In this paper we are concerned in the first instance with the type of normal elliptic scroll 1R22m+1[2m] whose points can be represented by the unordered pairs (u1; u2) of values of an elliptic parameter u(mod 2ω1, 2ω2); and we establish a significant connection between any normal parametrisation of the generators of 1R2m+1 and an associated parametric representation (u1u2) of its points. We also add a brief note to indicate the lines along which this kind of connection can be extended to apply to a general normal elliptic scrollar variety 1Rkmk+1[mk] whose points can be represented by the unordered sets (u1, …, uk) of values of an elliptic parameter u.

An ordered triangle in the plane S2 is defined as a sextuple (PlP2, P3; l1, l2, l3) consisting of three points Pi and three lines lj restricted by the relations of incidence Pi Ì lj (i ¹ j) If we map unrestricted sextuples by the points of a V12—Segre product of six planes—we obtain an image-manifold Ω6 for ordered triangles as the appropriate subvariety of V12. The variety Ω6 possesses an ordinary double threefold ɸ3 whose points map the totally degenerate triangles (i.e. those for which P1 ═ P2 ═ P3 and l1 ═ l2 ═ l3); Ω6 is therefore unsuitable as a basis for the construction of an enumerative calculus for triangles, for equivalence theory is as yet developed satisfactorily only on non-singular varieties.

We prove that the eigenvalues of a certain highly non-self-adjoint operator that arises in fluid mechanics correspond, up to scaling by a positive constant, to those of a self-adjoint operator with compact resolvent; hence there are infinitely many real eigenvalues which accumulate only at ±∞. We use this result to determine the asymptotic distribution of the eigenvalues.

A strong quantitative form of Manin’s conjecture is established for a certain variety in biprojective space. The singular integral in an application of the circle method involves the third power of the integral sine function and is evaluated in closed form.

The principal objective of this work is to investigate various classes of centrally symmetric convex sets. These classes range from the zonoids at one extreme to the class of all centrally symmetric bodies at the other. The defining properties of these classes involve inequalities between mixed volumes. Various other characterizations will be found in response to a number of questions in a recent survey article by Rolf Schneider and Wolfgang Weil. Some of these are concerned with measures on a Grassmannian manifold while others relate to the intermediate surface area measures of convex bodies. We shall also show these classes are characterized by certain extremal geometric inequalities. The work concludes with a brief discussion of related results concerned with generalized zonoids.

We prove that for every strictly convex body C in the Euclidean space of dimension d≥3, some aflfine image of C admits a non-lattice covering of the space, thinner than any lattice covering. We illustrate the general construction with an example of a thin non-lattice covering of with certain congruent ellipsoids.

The aim of this article is to present a general “large deviations approach” to the geometry of polytopes spanned by random points with independent coordinates. The origin of our work is in the study of the structure of ±1-polytopes, the convex hulls of subsets of the combinatorial cube . Understanding the complexity of this class of polytopes is important for the “polyhedral combinatorics” approach to combinatorial optimization, and was put forward by Ziegler in [20]. Many natural questions regarding the behaviour of ±1-polytopes in high dimensions are open, since, for many important geometric parameters, low-dimensional intuition does not help to identify the extremal ±1-polytopes. The study of random ±1-polytopes sheds light to some of these questions, the main reason being that random behaviour is often the extremal one.

Davies and Rogers [5] constructed a compact metric space Ω which is singular for a certain Hausdorff measure μh, in the sense that all subsets of Ω have μh-measure zero or infinity and μh(Ω) = ∞. (For a further study of this example see Boardman [3]). The interest lies in its extremely good descriptive character, which was lacking in the earlier examples given by Besicovitch [2] (a plane set singular for linear measure) and Choquet [4] (a plane set singular for any Hausdorff measure for which a segment has positive measure).

The results we present were motivated by the product measure problem for Baire measures. For two completely regular Hausdorff spaces X and Y, with totally finite a- additive measures μ and ν defined on the Baire σ- algebras ℬ0(X) and ℬ0(Y) respectively, under what conditions may we define a measure λ on the Baire σ-algebra ℬ0(X × Y), extending the product measure μ ⊗ ν defined on the product σ-algebra ℬ0(X) × ℬ0(Y) and satisfying a Fubini theorem?

Let x1, …, xn be linear forms in u1, …, un with real coefficients, and (for simplicity) determinant 1. Given a form (that is, a homogeneous polynomial) F(x1, …, xn), we can ask the following question: do there exist, for arbitrary real α1, …, αn, integers ul, …, un such that

where C is a suitable number independent of α1, …, αn and of the particular linear forms x1 …, xn? In two well-known cases this is true: namely when

and when

where 1 ≤ r ≤ n − 1.

Consider a real valued Morse function f on a C2 closed connected n-dimensional manifold M. It is proved that a suitable Riemannian metric exists on M, such that f is harmonic outside the set of critical points of f of index 0 and n. The proof is based on a result of Calabi [1], providing a criterion for a closed one-form on a closed connected manifold to be harmonic with respect to some Riemannian metric.

In this paper we continue the investigation begun in [6] concerning the number of solutions of the inequality

for almost all α (in the sense of Lebesgue measure on ℝ), where β is a given real number, , and both m and n are confined to sets of numbertheoretic interest. Our aim is to extend existing results ([7], [8], [5] for example), where only n is restricted. Here we shall prove the following result where, as elsewhere in this paper, p denotes a prime, and a square-free integer may be positive or negative.

Let R be a commutative ring with identity, and let U be a unitary commutative R-algebra with identity. In [1] Gilmer defines the (l/n)th power (n a positive integer) of a valuation ideal R when R is a domain. Sections 2, 3 of the present note are devoted to the study of an extension of this notion to positive rational powers of an arbitrary R-submodule of U.