§0. Einführung. Durch die bekannten Croftonschen Integrale können bekanntlich die Minkowskischen Quermaβintegrale konvexer Körper dargestellt werden. In der vorliegenden Note betrachten wir gewisse Erweiterungen dieser klassischen integralgeometrischen Formeln, durch die allgemeinere invariante Eikörperfunktionale gegeben sind. Es handelt sich hierbei um kinematische Integrale mit beweglichen unterdimensionalen Teilräumen, wobei passend gewahlte Funktionen ihrer Abstände vom Eikörper eingehen.

It is well known that two Borel subsets of the unit interval are Borel isomorphic, if, and only if, they have the same cardinality. The problem of the existence of analytic, non-Borelian subsets of the unit interval, which are not Borel isomorphic, has not been resolved within ZFC. With the additional assumption of the existence of an uncountable coanalytic set which does not contain a perfect set, it has been shown that there are at least three Borel isomorphism classes of analytic non-Borelian sets [4, 5].

Let w = f(z) be regular and schlicht for |z|, and f (0) = 0.

Suppose that f maps the unit disc {z : |z| < 1} onto a domain D starlike with respect to w = 0. Let C(r, θ) be the image in D of the ray joining z = 0 to z = reiθ, and let

be its length. Sheil–Small [1] proved that l(r, θ) < (1 + log 4) | f (reiθ)|, and conjectured the following result, which it is my aim to prove in this paper.

If an integer does not have a k-th power of a positive integer, other than 1, for a divisor, it is said to be k–free. Let f(n) be an irreducible polynomial, with rational integer coefficients, of degree g, having no fixed k-th power divisors other than 1. We define

i.e. Nk(x) is the number of positive integers n not exceeding x such that f(n) is k-free. One would expect that f(n) is square-free for infinitely many n and further that, given x sufficiently large, there is an n with x < n ≤ x + h, such that f(n) is square-free for h = 0(x2) where ε is any real number > 0. These conjectures, however, seem to be extraordinarily difficult to prove. We begin with a brief account of the best results that have been attained so far.

In a compact group G, a sequence (Fn) of finite sets is uniformly distributed if the averaging operators

are uniformly convergent to the mean for continuous complex-valued functions f. In any compact metric group, there are uniformly distributed sequences of finite sets which are determined by a metric for the group. In some compact groups, there are uniformly distributed sequences of finite sets which are determined by the algebraic structure. A necessary and sufficient condition for a sequence of finite sets to be uniformly distributed in a compact metric group is that for any metric d for G and each εG, there is a sequence of one-to-one maps pn: Fn→ Fn such that

In [1], P. X. Gallagher introduced a new sieve which is designed to produce better estimates than the large sieve when a vast number of congruence classes are chosen for each of the sieving primes. By making more explicit use of the principles underlying the sieve, these estimates can be improved and generalized to the case of complex quadratic fields. We shall see that the resulting estimates are best possible, if there are only a bounded number of unsieved classes per prime.

In this paper we are interested in two related measures of the degree of approximation of a complex number ζ by algebraic numbers. For a given integer n ≥ 1, write wn(ζ) for the supremum of the exponents w for which

for infinitely many polynomials

in Z[x] of height H(P) = max |av|. Clearly 0 ≤ w1 (ζ) ≤ w2 (ζ) ≤ …. On the other hand, write for the supremum of the exponents w for which

for infinitely many algebraic numbers α of degree at most n.