Recently, it has been shown that tight or almost tight upper bounds for the discrepancy of many geometrically denned set systems can be derived from simple combinatorial parameters of these set systems. Namely, if the primal shatter function of a set system ℛ on an n-point set X is bounded by const. md, then the discrepancy disc (ℛ) = O(n(d−1)/2d) (which is known to be tight), and if the dual shatter function is bounded by const. md, then disc We prove that for d = 2, 3, the latter bound also cannot be improved in general. We also show that bounds on the shatter functions alone do not imply the average (L1)-discrepancy to be much smaller than the maximum discrepancy: this contrasts results of Beck and Chen for certain geometric cases. In the proof we give a construction of a certain asymptotically extremal bipartite graph, which may be of independent interest.

In this paper, we consider the so-called “Furstenberg set problem” in high dimensions. First, following Wolff’s work on the two-dimensional real case, we provide “reasonable” upper bounds for the problem for $\mathbb{R}$ or $\mathbb{F}_{p}$. Next we study the “critical” case and improve the “trivial” exponent by ${\rm\Omega}(1/n^{2})$ for $\mathbb{F}_{p}^{n}$. Our key tool in obtaining this lower bound is a theorem about how things behave when the Loomis–Whitney inequality is nearly sharp, as it helps us to reduce the problem to dimension two.

We study the distribution of the size of Selmer groups and Tate–Shafarevich groups arising from a 2-isogeny and its dual 2-isogeny for elliptic curves En:y2=x3−n3. We show that the 2-ranks of these groups all follow the same distribution. The result also implies that the mean value of the 2-rank of the corresponding Tate–Shafarevich groups for square-free positive integers n≤X is as X→∞. This is quite different from quadratic twists of elliptic curves with full 2-torsion points over ℚ [M. Xiong and A. Zaharescu, Distribution of Selmer groups of quadratic twists of a family of elliptic curves. Adv. Math. 219 (2008), 523–553], where one Tate–Shafarevich group is almost always trivial while the other is much larger.

In this paper various analytic techniques are combined in order to study the average of a product of a Hecke $L$-function and a symmetric square $L$-function at the central point in the weight aspect. The evaluation of the second main term relies on the theory of Maaß forms of half-integral weight and the Rankin–Selberg method. The error terms are bounded using the Liouville–Green approximation.

Paucity is established for a system of diagonal diophantine equations, in which the degrees are the odd numbers in ascending order.

Fifty years ago Marcinkiewicz and Zygmund studied the circular structure of the limit points of the partial sums for (C, 1) summable Taylor series. More specifically, let

be a power series with complex coefficients, let

be the partial sums, and let

be the Cesàro averages. When the sequence σn(z) converges to a finite limit σ(Z), we say that the Taylor series is (C, 1) summable and σ(z) is the (C, 1) sum of the series. Concerning (C, 1) summable Taylor series Marcinkiewicz and Zygmund ([5], [6] Vol. II, p. 178) established the following theorem.

It has been shown in [1, 2] that it is sometimes possible to justify the uncoupled and quasi-static approximations which are commonly invoked to simplify the solution of initial and boundary value problems in the linear theory of thermoelasticity. The justification involves showing, among other things, that the temperature predicted by the coupled dynamic theory is approximated by a solution of the classical heat equation.

It is shown that every compact convex set in with mean width equal to that of a line segment of length 2 and with Steiner point at the origin is contained in the unit ball. As a consequence, the diameter with respect to the Hausdorff metric of the space of all such sets is 1. There also results a sharp bound for the Hausdorff distance between any two compact convex sets.

The reviewer [Math. Reviews, 32 (1966), 7526] has stated that he could not follow case (iii) of the proof of Lemma 7 and upon examination we find that the argument is incomplete (on p. 183, the degree of d1 may not be smaller than that of g). The following version of the proof (which is given with full details), circumvents the difficulty and has been designed to show, further, that the constant implied in the O-symbol is independent of the number of variables. This minor refinement may be of some interest as the corresponding estimates of Lang and Weil [Amer. J. Math., 76 (1954), 819–827; cf., Lemmas 1, 2 and Theorem 1] for general varieties over a finite field lack this feature. Throughout, we use the symbol deg F to denote the total degree of an element F of a polynomial domain , where k = [p] is the field of residue classes mod p and . Nr(†) will denote the number of r-tuples (xn−r+1, …, xn) ε kr with some specific property †. We prove that, if n ≥ 1 and f, g are elements of with no non-constant common factor in , i.e. (f, g)p = 1, then

where the constant in the O-symbol depends only upon d = max (deg f, deg g). The argument is entirely elementary and uses a method of descent in which the number of variables n and the minimum degree, defined by

are simultaneously diminished at each step of the descent. It makes use of a corresponding (trivial) estimate for the case of one polynomial, i.e. if and ∂ = deg F, then Nn(F = 0) = O∂(pn−1). Thus, if f, g are disjoint in the sense that f ε k[x1, …, xr], g ε k[xr+1, …, xn] after a suitable permutation of x1, …, xn, then (f, g)p = 1 implies that

Let 2 = p1 < p2 < … be the sequence of consecutive prime numbers. Put dn = pn+l − pn. Turán and I proved [1] that the inequalities dn+1 > dn and dn+1 < dn both have infinitely many solutions. It is not known if dn = dn+l has infinitely many solutions. The answer is undoubtedly affirmative but the proof will probably be very difficult [2]. It was a great surprise and disappointment to us that we could not prove that dn+2 > dn+1 > dn has infinitely many solutions. We could not even prove that (− 1)n (dn+l − dn) changes sign infinitely often. It seems certain that the answer to both of these questions is affirmative and perhaps a simple proof can be found.

Some sixty years ago Hardy and Ramanujan [6]introduced the notion of normal order of an arithmetic function.

A real-valued arithmetic function f)n) is said to have a normal order if there is a function g(n), which is non-negative and non-decreasing for all sufficiently large integers n, so that, for each fixed ε > 0, the integers n, for which the inequality

is satisfied, have asymptotic density zero. Thus, in this certain sense, f(n) behaves almost surely like g(n). We say that f(n) has the normal order g(n). In their original paper Hardy and Ramanujan asked that the function g(n) be “elementary”, but this is a requirement that subsequent researchers have dropped.

§1. Introduction and main results. A map f: A → R (A ⊂ R) is called piecewise contractive if there is a finite partition A = A1∪ … ∪ An such that the restriction f| Ai is a contraction for every i = 1, …, n. According to a theorem proved by von Neumann in [3], every interval can be mapped, using a piecewise contractive map, onto a longer interval. This easily implies that whenever A, B are bounded subsets of R with nonempty interior, then A can be mapped, using a piecewise contractive map, onto B (see [6], Theorem 7.12, p. 105). Our aim is to determine the range of the Lebesgue measure of B, supposing that the number of pieces in the partition of A is given. The Lebesgue outer measure will be denoted by λ. If I is an interval then we write |I| = λ(I).

For satellite knots there is a well-known formula which relates the Alexander polynomial of the satellite to those of a companion knot and the corresponding pattern. If &s, &C and &P are the Alexander polynomials of a satellite, companion and pattern respectively then

where is the linking number of P with a meridian of the companion torus (see [BZ], p. 118). Analogous relationships do not exist for other knot polynomials [MS]. This suggests that the existence of the above formula depends more on the geometry underlying the polynomial than on the geometry of the satellite construction.

In line with the Ritt–Seidenberg elimination theorem in differential algebra [RIT], [SEI], and with an “approximation theorem” by Denef and Lipshitz [DEL] for formal power series, and with an elimination theorem by the author [RUB1] for C∞ solutions of systems of algebraic differential equations (ADE's), one is led to consider the corresponding elimination question for Cn solutions. Somewhat in the spirit of [RUB2], though, we produce a negative result in this direction.

Methods are used from descriptive set theory to derive Fubinilike results for the very general Method I and Method II (outer) measure constructions. Such constructions, which often lead to non-σ-finite measures, include Carathéodory and Hausdorff-type measures. Several questions of independent interest are encountered, such as the measurability of measures of sections of sets, the decomposition of sets into subsets with good sectional properties, and the analyticity of certain operators over sets. Applications are indicated to Hausdorff and generalized Hausdorff measures and to packing dimensions.

§1. Introduction. We study in this paper some properties of the Lusternik-Schnirelmann category of isolated invariant sets of continuous dynamical systems. There are several different definitions of this coefficient, although most of them agree in the important case of ANR's (Absolute Neighbourhood Retracts). We refer to the review articles [10] by R. H. Fox and [15, 16] by I. M. James for general information about this topological invariant. We shall use in this paper the definition of the Lusternik-Schnirelmann category of a compactum introduced by K. Borsuk in [4].

A (local) Lie loop is a real analytic manifold M with a base point e and three analytic functions (x, y) → x° y, x\y, x/y: M × M → M (respectively, U × U → M for an open neighbourhood U of e in M) such that the following conditions are satisfied: (i) x ° e = e ° x = e, (ii) x ° (x\y) = y, and (iii) (x/y)° y = x for all x, y ε M (respectively, U). If the multiplication ° is associative, then M is a (local) Lie group. The tangent vector space L(M) in e is equipped with an anticommutative bilinear operation (X, Y) →[X, Y] and a trilinear operation (X, Y, Z) →〈X, Y, Z〉. These are defined as follows: Let B be a convex symmetric open neighbourhood of 0 in L(M) such that the exponential function maps B diffeomorphically onto an open neighbourhood V of e in M and transport the operation ° into L(M) by defining X ° Y = (exp|B)−1((exp X)° (exp Y)) for X and Y in a neighbourhood C of 0 in B such that (exp C) ° (exp C) ⊂ V. Similarly, we transport / and \.

Let X ⊂ ℙN be a geometrically integral cubic hypersurface defined over ℚ, with singular locus of dimension at most dim X − 4. The main result in this paper is a proof of the fact that X(ℚ) contains OɛX (BdimX + ɛ) points of height at most B.

We consider exponential sums with x-coordinates of points qG and q−1G where G is a point of order T on an elliptic curve modulo a prime p and q runs through all primes up to N (with gcd (q,T)=1 in the case of the points q−1G). We obtain a new bound on exponential sums with q−1G and correct an imprecision in the work of W. D. Banks, J. B. Friedlander, M. Z. Garaev and I. E. Shparlinski on exponential sums with qG. We also note that similar sums with g1/q for an integer g with gcd (g,p)=1 have been estimated by J. Bourgain and I. E. Shparlinski.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{F}_q$ be the finite field of $q$ elements. An analogue of the regular continued fraction expansion for an element $\alpha $ in the field of formal Laurent series over $\mathbb{F}_q$ is given uniquely by

$$\begin{equation*} \alpha = A_0(\alpha )+\cfrac {1}{A_1(\alpha )+\cfrac {1}{A_2(\alpha )+\ddots }}, \end{equation*}$$

$(A_n(\alpha ))_{n=0}^\infty $

$\mathbb{F}_q$

$\deg (A_n(\alpha ))\ge 1$

$n\ge 1.$

$(p_n)_{n=1}^\infty $

$$\begin{equation*} \lim _{n\to \infty }\frac {1}{n}\sum _{j=1}^n \deg (A_{p_j}(\alpha )) = \frac {q}{q-1} \end{equation*}$$

$\alpha $

$(p_n)_{n=1}^\infty $

$(n)_{n=1}^\infty ,$