A line that intersects every member of a finite family F of convex sets in the plane is called a common transversal to F. In this paper we study some basic properties of T(k)-families: finite families of convex sets in the plane in which every subfamily of size at most k admits a common transversal. It is known that a T(k)-family admits a partial transversal of size α∣F∣ for some constant α(k) which is independent of F. Here it will be shown that (2/(k(k−1)))1/(k−2)≤α(k)≤((k−2)/(k−1)), which are the best bounds to date.

We prove a family of sharp bilinear space–time estimates for the half-wave propagator $\text{e}^{\text{i}t\sqrt{-\unicode[STIX]{x1D6E5}}}$ . As a consequence, for radially symmetric initial data, we establish sharp estimates of this kind for a range of exponents beyond the classical range.

We obtain density estimates for subsets of the $n$-dimensional integer lattice lacking four-term matrix progressions. As a consequence, we show that a subset of the grid $\{1,2,\dots ,N\}^{2}$ lacking four corners in a square has size at most $\mathit{CN}^{2}(\log \log N)^{-c}$. Our proofs involve the density increment method of Roth [J. London Math. Soc.28 (1953), 104–109] and Gowers [Geom. Funct. Anal.11(3) (2001), 465–588], together with the $U^{3}$-inverse theorem of Green and Tao [Proc. Edinb. Math. Soc. (2) 51(1) (2008), 73–153].

In this paper we prove sharp estimates for generalized eigenvectors of Hermitian Jacobi matrices associated with the spectral parameter lying in a gap of their essential spectra. The estimates do not depend on the main diagonals of these matrices. The types of estimates obtained for bounded and unbounded gaps are different. These estimates extend the previous ones found in [J. Janas, S. Naboko and G. Stolz, Decay bounds on eigenfunctions and the singular spectrum of unbounded Jacobi matrices. Int. Math. Res. Not.4 (2009), 736–764], for the spectral parameter being outside the whole spectrum of Jacobi matrices. Examples illustrating optimality of our results are also given.

Starting with a paper of Jacobson from the 1960s, many authors became interested in characterizing all algebraic number fields in which each integer is the sum of pairwise distinct units. Although there exist many partial results for number fields of low degree, a full characterization of these number fields is still not available. Narkiewicz and Jarden posed an analogous question for sums of units that are not necessarily distinct. In this paper we propose a generalization of these problems. In particular, for a given rational integer n we consider the following problem. Characterize all number fields for which every integer is a linear combination of finitely many units εi in a way that the coefficients ai∈ℕ are bounded by n. The paper gives several partial results on this problem. In our proofs we exploit the fact that these representations are related to symmetric beta expansions with respect to Pisot bases.

We investigate the approximation of quadratic Dirichlet $L$-functions over function fields by truncations of their Euler products. We first establish representations for such $L$-functions as products over prime polynomials times products over their zeros. This is the hybrid formula in function fields. We then prove that partial Euler products are good approximations of an $L$-function away from its zeros and that, when the length of the product tends to infinity, we recover the original $L$-function. We also obtain explicit expressions for the arguments of quadratic Dirichlet $L$-functions over function fields and for the arguments of their partial Euler products. In the second part of the paper we construct, for each quadratic Dirichlet $L$-function over a function field, an auxiliary function based on the approximate functional equation that equals the $L$-function on the critical line. We also construct a parametrized family of approximations of these auxiliary functions and prove that the Riemann hypothesis holds for them and that their zeros are related to those of the associated $L$-function. Finally, we estimate the counting function for the zeros of this family of approximations, show that these zeros cluster near those of the associated $L$-function, and that, when the parameter is not too large, almost all the zeros of the approximations are simple.

We study the problem of Diophantine approximation on lines in $\mathbb{R}^{d}$ under certain primality restrictions.

This paper is concerned with (a) a new simple method of solution of a wide variety of problems of elastic strips by means of Fourier transforms in the complex plane and (b) a direct solution of the elastic annulus. Continuation of functions into adjacent regions of the plane and the solution of differential-difference equations are seen to be unnecessary complications.

§1. Introduction. In [1], Odoni discusses the iterates of the polynomial x2 +1 and their Galois groups over the rationals (a problem initially proposed by J. McKay). Setting f1,(x) = x2+1 and fn(x) =f1(fn-1(X)) for n ≥ 2, write Kn for the splitting field of fn(x) over and Ωn = Gal (Kn/). Then Odoni proves that Ωn is isomorphic to a subgroup of [C2]n, the nth wreath power of the cyclic group C2 of order 2, and gives a simple rational criterion for Ωn [C2]n to hold. In this note we describe a computer implementation of Odoni's criterion, and state the result that Ωn [C2]n for n ≤ 5 × 107.

Questions of Haight and of Weizsäcker are answered in the following result. There exists a measurable function f: (0, + ∞) → {0,1} and two non-empty intervals IFI∞⊂[½,1) such that Σ∞n = 1f(nx) = +∞ for everyx εI∞, and Σ∞n = 1f(nx) >+∞ for almost every xεIf. The function f may be taken to be the characteristic function of an open set E.

If f(s) is the analytic function defined by the Dirichlet series and if where 0 ≤ b < 1, then the series converges for Re s > 1 and f(s) is regular in the half plane Re s > b except for a simple pole with residue C ≠ 0 at a s = 1. Thus f(s) has a Laurent expansion at s = 1 and it has been shown [1] that under these conditions

where

Let (Sn)n = 1,2,… be the strictly increasing sequence of those natural numbers that can be represented as the sum of three cubes of positive integers. The estimate

is easily proved as follows: Let x1 be the largest natural number with Then This procedure is iterated by choosing x2 and then x3 as the largest natural numbers satisfying and Thus Since this implies (1).

In a recent paper [4], I introduced the notion of recursive formal Lie groups (of infinite dimension) over a field of characteristic p > 0, and studied a particular class of such groups, the groups of hyperexponential type; these can be characterized either as being (recursively) isomorphic to a special group of that class, the hyperexponential group, or by simple conditions on their Lie hyperalgebra. An interesting example of a group of that class is the additive Witt group W, whose “infinitesimal” structure can therefore be considered as known, at least “up to an isomorphism”. However, the intrinsic importance of the Witt group (which, as well known, is the “formalization”, so to speak, of the additive group of a p-adic field) leads one to think that it may be worth while to study in greater detail that group itself, instead of being content with the mere existence of an unspecified isomorphism with the hyperexponential group. This is what we intend to do here; it turns out that, although it seems hopeless to write down explicitly the group law of the Witt group, the multiplication table of its hyperalgebra is, on the contrary, as simple as one could hope, and is, in fact, identical to that of the hyperalgebra of the hyperexponential group (although the two groups are distinct). Moreover, this leads to a new and quite unexpected definition of the Witt group, which links it still closer to the hyperexponential group, and provides a well-determined isomorphism between the two groups.

In this paper the absolute value or distance from the origin analogue of the classical Khintchine-Groshev theorem [5] is established for a single linear form with a “slowly decreasing” error function. To explain this in more detail, some notation is introduced. Throughout this paper, m, n are positive integers; i.e., m, n ∈ ℕ; x = (x1,…, xn) will denote a point or vector in ℝn, q = (q1,…, qn) will denote a non-zero vector in ℤn and

|x| := max{|x1|,…,|xn|} = ‖X‖∞

will denote the height of the vector x. Let Ψ : ℕ → (0, ∞) be a (non-zero) function which converges to 0 at ∞. The notion of a slowly decreasing functionΨ is defined in [3] as a function for which, given c ∈ (0, 1), there exists a K = K(c) > 1 such that Ψ(ck) ≤ KΨ(k). Of course, since Ψ is decreasing, Ψ(k) ≤ Ψ(ck). For any set X, |X| will denote the Lebesgue measure of X (there should be no confusion with the height of a vector).

The Dedekind η-function is defined by

and is well-known to satisfy a functional equation relating η(aτ + b/cτ + d) to η (τ), where a, b, c, d are rational integers with ad – bc = 1—see for instance Iseki [2], and the further references cited there.

We shall be concerned with two boundary value problems for the Falkner-Skan Equation

when –β is a small positive number. Our interest is in solutions of (1) which exhibit “reversed flow”; that is, solutions f such that f′(x) < 0 for small positive values of x. The boundary conditions which we wish to consider are

and

One of the converse statements to Lagrange's theorem is that, for each subgroup H of G and any prime factor p of |G: H|, there exists a subgroup K such that H≤K≤G with |K: H | = p. This paper treats integers n such that all groups of order n have this property.

Let μ be a Borel measure on a completely regular space X, and denote by ℱ the σ-algebra of all μ*-measurable subsets of X. Suppose that, as an abstract measure space, (X, ℱ, μ) is isomorphism mod zero with the standard Lebesgue space (I, ℒ, m) via an isomorphism φ : X → I. In this note we attempt to answer the following question: Under what conditions can the isomorphism φ be chosen to be a homeomorphism mod zero? When X is compact, the existence of such a homeomorphism was established in [3, §4] under the assumption of uniform regularity of μ. Whether or not the result can be established without this assumption, was posed as an open question there. Here, we give necessary and sufficient conditions for the existence of the above homeomorphism, together with various examples showing, among other things, that the assumption of uniform regularity used in [3, §4] cannot be dropped.

We study infima of families of topologies on the hyperspace of a metrizable space. We prove that Kuratowski convergence is the infimum, in the lattice of convergences, of all Wijsman topologies and that the cocompact topology on a metric space which is complete for a metric d is the infimum of the upper Wijsman topologies arising from metrics that are uniformly equivalent to d.

Heilbronn [5] proved that for any ε > 0 there exists C(ε) such that for any real θ and N ≥ 1 there is an integer x satisfying

where ‖α‖ denotes the difference between α and the nearest integer, taken positively. The result is uniform in θ and so analogous to Dirichlet's inequality for the fractional parts of nθ. The result has been generalized to simultaneous approximations by Danicic [1] and Ming-chit Liu [6]. Here we shall extend the result to any finite number of simultaneous approximations when x2 is replaced by a k-th power.