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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 e i in a way that the coefficients a i ∈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.

Recently, Steinerberger proved a uniform inequality for the Laplacian serving as a counterpoint to the standard uniform sublevel set inequality which is known to fail for the Laplacian. In this paper, we observe that many inequalities of this type follow from a uniform lower bound on the $L^1$ norm, and give an analogous result for any linear differential operator, which can fail for non-linear operators. We consider lower bounds on the $L^p$ quasi-norms for $p, 42 pages, 1 figure, supersedes arXiv:2005.09407: main theorems strengthened and surrounding discussion presented in a much different context

In recent papers on fractals attention has shifted from sets to measures [1, 5, 10]. Thus it seems interesting to know whether results for the dimension of sets remain valid for the dimension of measures. In the present paper we derive estimates for the dimension of product measures. Falconer [3] summarizes known results for sets and Tricot [8] gives a complete description in terms of Hausdorff and packing dimension. Let dim and Dim denote Hausdorff and packing dimension. If then

The paper is devoted to the study of Fefferman–Stein inequalities for stochastic integrals. If is a martingale, is the stochastic integral, with respect to , of some predictable process taking values in , then for any weight belonging to the class we have the estimates and The proofs rest on the Bellman function method: the inequalities are deduced from the existence of certain special functions, enjoying appropriate majorization and concavity. As an application, related statements for Haar multipliers are indicated. The above estimates can be regarded as probabilistic counterparts of the recent results of Lerner, Ombrosi and Perez concerning singular integral operators.

The main new difficulties encountered are the possible existence of inseparable elements in the finite extension ft/P, and also the occurrence of constant field extensions. The former phenomenon is overcome in §1 by the device of replacing P by a suitable purely inseparable extension K s fi for which " descent by norms " to P behaves in a reasonable manner. We then attempt to imitate where possible the processes of [1] applied to Q.JK. The presence of constant field extensions inevitably complicates the analysis, since the " obvious " analogue of Cebotarev's density theorem is unexpectedly false in this case. We append to this paper a section on the appropriate Cebotarev theorem which, we hope, will fill the gap in the literature of this intriguing topic. Section 2 of this paper corresponds broadly to §1 of [1], and in it we prove THEOREM 1IA. Let G¥(q s ) be the constant field ofd, a finite extension of P. Then the number of integral ideals ofk[X] which have degree d, are prime to a given ideal n, and are norms of fractional ideals of Cl is zero unless f divides d, in which case the number is asymptotically

Motivated by questions of computational complexity, Rabin [7] introduced the notion of a complete proof of a system of inequalities. His work and the related paper of Spira [8] should interest geometers as well as computer scientists, for both papers involve convexity in an essential way. Spira's results concern the possibility of covering the intersection of a convex set C and a convex polyhedron Q with a finite collection P of polyhedra subject to certain conditions, while in Rabin's work the members of P may be more general than polyhedra. Both papers are interesting and treat important questions, but only Rabin's paper is correct in all respects. The present note contains counterexamples to some of Spira's results and establishes a correct version of one of them.

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.

It was proved by Roth in a recent paper that if α is any real algebraic number, and if K > 2, then the inequalityhas only a finite number of solutions in relatively prime integers p, q (q > 0) The object of the present paper is to prove that the lower bound for κ can be reduced if conditions are imposed on p and q. The result obtained is as follows.

In this paper we generalize and extend work by Languasco, Perreli and Zaccagnini on the Montgomery–Hooley theorem in short intervals. We are thus able to obtain asymptotic formulae for the mean-square error in the distribution of general sequences in arithmetic progressions in short intervals. As applications we consider lower and upper bound approximations to the characteristic function of the primes in a short interval and an average over short intervals for the von Mangoldt function.

The following conjecture generalizing the Contraction Mapping Theorem was made by Stein in [9]: Let (X, ρ) be a complete metric space and let F = {T1, . . . , Tn} be a finite family of self-maps of X. Suppose there is a constant γ ∈ (0, 1) such that for any x, y ∈ X there exists T ∈ F with ρ(T (x), T (y)) ≤ γρ(x, y). Then some composition of members of F has a fixed point. In this paper we disprove this conjecture. We also show that it does hold for a (continuous) commuting F in the case n = 2. We conjecture that it holds for commuting F for any n.

For the purpose of this paper, we call a set of positive reals ν a well-spaced set if there is a c > 0 such that

Let υ be a Henselian valuation of any rank of a field K and υ its unique prolongation to a fixed algebraic closure K of K having value group G. For any subfield L of K, let R(L) denote the residue field of the valuation obtained by restricting υ to L. Using the canonical homomorphism from the valuation ring of υ onto its residue field R(K), one can lift any monic irreducible polynomial with coefficients in R(K) to yield a monic irreducible polynomial with coefficients in K. In an attempt to generalize this concept, Popescu and Zaharescu introduced the notion of lifting with respect to a (K, υ)-minimal pair (α, δ) belonging to K × G. As in the case of usual lifting, a given monic irreducible polynomial Q(y) belonging to R(K(α))[y] gives rise to several monic irreducible polynomials over K which are obtained by lifting with respect to a fixed (K, υ)-minimal pair (α, δ). If F, F 1 are two such lifted polynomials with coefficients in K having roots θ, θ 1 , respectively, then it is proved in the present paper that υ(K(θ)) = υ(K(θ 1 )), R(K(θ)) = R(K(θ 1 )); in case (K, υ) is a tame field, it is shown that K(θ) and K(θ 1 ) are indeed K-isomorphic.

In this paper, further insight is obtained into the earlier approach of studying residually transcendental extensions of a valuation v of a field K to a simple transcendental extension K(x) of K by means of minimal pairs, thereby introducing new invariants corresponding to any element of an algebraic closure λ of K. It is also shown that these invariants are of independent interest as well. A characterization of those elements a belonging to λ is given such that there exists a minimal pair (a, δ) for some δ in the divisible closure of the value group of v.

A well-known result of Ehrenfeucht states that a difference polynomial f(X)-g(Y) in two variables X, Y with complex coefficients is irreducible if the degrees of f and g are coprime. Panaitopol and Stefanescu generalized this result, by giving an irreducibility condition for a larger class of polynomials called “generalized difference polynomials”. This paper gives an irreducibility criterion for more general polynomials, of which the criterion of Panaitopol and Stefanescu is a special case.

A universal algebra is called congruence compact if every family of congruence classes with the finite intersection property has a non-empty intersection. This paper determines the structure of all right congruence compact monoids S for which Green's relations ℐ and ℋ coincide. The results are thus sufficiently general to describe, in particular, all congruence compact commutative monoids and all right congruence compact Clifford inverse monoids.

It is known that a system of r additive equations of degree k with greater than 2 rk variables has a non-trivial p -adic solution for all p > k 2 r + 2 . In this paper we consider the same system with more than crk variables, c > 2, and show the existence of a non-trivial solution for all p > r 2 k 2+(2/( c − 2)) if r ≠ 1 and p > k 2+(2/( c − 1)) if r = 1.

Rabinowitz' global bifurcation theorem shows that for a large class of nonlinear eigenvalue problems a continuum ( i.e ., a closed, connected set) of solutions bifurcates from the trivial solution at each eigenvalue (or characteristic value) of odd multiplicity of the linearized problem (linearized at the trivial solution). Each continuum must either be unbounded, or must meet some other eigenvalue. This paper considers a class of such nonlinear eigenvalue problems having simple eigenvalues and a “weak” nonlinear term. A result regarding the location of the continua is obtained which shows, in particular, that in this case the bifurcating continua must be unbounded. Also, under further differentiability conditions it is shown that the continua are smooth, 1-dimensional curves and that there are no non-trivial solutions of the equation other than those lying on the bifurcating continua.

Let ν be a valuation of any rank of a field K with value group G ν and f(X)= X m + a l X m−1 + … + a m be a polynomial over K. In this paper, it is shown that if (ν(a i )/i)≥(ν(a m )/m)>0 for l≤i≤m, and there does not exist any integer r>1 dividing m such that ν(a m )/r∈G ν , then f(X) is irreducible over K. It is derived as a special case of a more general result proved here. It generalizes the usual Eisenstein Irreducibility Criterion and an Irreducibility Criterion due to Popescu and Zaharescu for discrete, rank-1 valued fields, (cf. [Journal of Number Theory, 52 (1995), 98–118]).

In this paper we study various classes of centrally symmetric sets in d-dimensional Euclidean space Rd. As we will see, it is appropriate to focus our attention on those sets which have interior points.

Let λ1, …, λs be nonzero real numbers and suppose that λ1/λs is irrational. In 1955, Davenport and Roth showed [6] that the values taken byat integer points (x1, …, xs) are dense on the real line, provided that s≥8. In the present paper we obtain the same result with seven variables.

Let g(n) be a complex valued multiplicative function such that |g(n)| ≤ 1. In this paper we shall be concerned with the validity of the inequalityunder the weak condition g(p)∈ for all primes p, where is a fixed subset of the closed unit disc Thus our point of view is similar to that of Halasz [Hz 2] in that we seek a general inequality in terms of simple quantities, albeit g(p) may have a quite irregular distribution. We are not concerned here with the problem of asymptotic formulae for the sum on the left of (1) studied by (among others) Delange [D], Halasz [Hz 1] and Wirsing [W].

In this paper, we first give a characterization of a regular endomorphism of a graph. Then, we explicitly describe its inverses through idempotents.

A compact space K is said to have the Namioka Property , or to belong to the class *, if, for every Baire space B and every separately continuous function Ψ: B × K → ℝ, there is dense δ subset H of B such that Ψ is (jointly) continuous at all points of H × K . Although the terminology is more recent, the idea of looking at properties of this kind goes back to Namioka's paper [6] on separate and joint continuity. Talagrand [8] gave the first example of a compact space that is not in * and it is now known [4] that there are even examples of scattered compact spaces that are not in *. On the other hand, many good classes of compact spaces have been shown to be contained in *, probably the most general being the class of continuous images of Valdivia compacts [2]. The aim of this note is to prove the following stability result: a compact space which is a countable union of closed subsets with the Namioka Property does itself possess that property.

The object of this paper is to obtain improvements in Vinogradov's mean value theorem widely applicable in additive number theory. Let Js,k(P) denote the number of solutions of the simultaneous diophantine equationswith 1 ≥ xi, yi ≥ P for 1 ≥ i ≥ s. In the mid-thirties Vinogradov developed a new method (now known as Vinogradov's mean value theorem) which enabled him to obtain fairly strong bounds for Js,k(P). On writingin which e(α) denotes e2πiα, we observe thatwhere Tk denotes the k-dimensional unit cube, and α = (α1,…,αk).

§1. Introduction. In this paper we give a new proof of a theorem of Bombieri and Davenport [2, Theorem 1]. Let t(–k) = t(k) be real,where e(u) = e2πiu. Let p and p' denote primes, k an integer, and define

Let V 0 be a discrete real valuation of a field K and x an indeterminate. In 1936, MacLane [3] gave a method of constructing all real valuations of K(x) which are extensions of V 0 .In this paper, we determine explicitly all rank 2 valuations of K(x) which extend V 0 .One can thereby describe all rank 2 valuations of K(x, y) which are trivial on an arbitrary K; x, y being algebraically independent over the field K. The latter valuations have been considered by Zariski [5] in the case when K is an algebraically closed field of characteristic zero.

Let k be a knot of type K and with group πK. Let θ: nK → PSL(ℂ) = PSL (2, ℂ) be a parabolic representation (p-rep) as defined in [14]. We shall call the representation discrete when its image πKθ is a discrete subgroup of PSL(ℂ). It is known that PSL(ℂ) can be identified with the group of orientation preserving isometries of hyperbolic 3-space , and that each discrete subgroup of PSL(ℂ) acts discontinuously on . Hence each discrete p-rep θ has an associated orbit space . The present paper is a study of the general relations between the algebraic properties of a discrete image πKθ and the geometric properties of its orbit space.

§1. Preliminaries. Let X be a stable subordinator in R, B a subset of R and A a time set. In this paper we shall consider the Hausdorff dimension properties of the random sets

Let g ( n ) be a complex-valued multiplicative function which satisfies | g ( n )| ≤ 1 for all n. The aim of this paper is to establish the following theorem.

Let K be a number field and E/K an elliptic curve. As is well known [3, 4,[ if K has class number 1, then there exists a global minimal Weierstrass equation for E. Our main goal in this paper is to prove the following converse to this statement.

The question, whether a given element of a C*-algebra has an image of rank one in some faithful representation, was studied in [3]. Such elements were characterised there by the property of being “single” (as defined below). As was pointed out in [3], Section 5, this criterion fails for general Banach algebras and the purpose of this paper is to provide a stronger condition giving the required representation property for any semi-simple Banach algebra.

Throughout this paper A is a commutative noetherian ring (with identity) and M is an A -module. We use to denote, for i ≥ 0, the i -th right derived functor of the local cohomology functor L with respect to an ideal a of A [8; 2.1].

Let G be any enumerable subset of the positive real numbers, with infinity as its only limit point. The purpose of this paper is to give a construction for a Lebesgue measurable set E ⊂ R+, with the following properties

In this paper is proved a formula for the number of neighbourly d-polytopes with d + 3 vertices, when d is odd.

This paper gives an algorithm for generating all the solutions in integers x0, x1…, xn of the inequalitywhere 1, α1, …,αn are numbers, linearly independent over the rationals, in a real algebraic number field of degree n + 1 ≥ 3 and c is any sufficiently large positive constant. It is well known [2, p. 79] that if c is small enough, then (1) has no integer solutions with x1…, xn not all zero.

In this paper I show that complex-valued multiplicative functions g which satisfy |g(n)|≤1 for all positive integers n, are generally well distributed in residue classes to small moduli.

This paper deals with the problem (raised by J. Browkin) of how many ring generators are needed for the ring of integers of a given algebraic number field. I show that the number of generators needed for the integers of a field of degree n is less than (log n /log2) + 1, and that if 2 splits completely in the field the number of generators needed is in fact the largest integer less than (log n /log2) + 1. These results follow from a computable formula (that depends only on how the small primes factorize in the field) for the number of generators of the ring of integers of a given field. This formula has the single drawback that when it yields “one” two generators may be needed, and I show that there are fields of arbitrarily high degree for which this happens.

In this paper we give alternative proofs of some formulas of Shintani [1, 2] and Zagier [3] concerning Kronecker limit formulas for real quadratic fields.

We write e(x) for e2πix and let ‖x‖ denote the distance of x from the nearest integer. The notation A ≪ B will mean |A| ≤ C|B| where C is a positive constant depending at most on an arbitrary positive number e, and on an integer k. The letter p always denotes a prime number. The main results of the present paper are as follows.

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 whichfor infinitely many polynomialsin 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 whichfor infinitely many algebraic numbers α of degree at most n.

A d-dimensional zonotope Z in Ed which is the vector sum of n line segments is linearly equivalent to the image of a regular n-cube under some orthogonal projection. The zonotope S0025579300006082_inline1 in En-d which is the image of the same cube under projection on to the orthogonal complementary subspace is said to be associated with Z. In this paper is proved a conjecture of G. C. Shephard, which asserts that, if Z tiles Ed by translation, with adjacent zonotopes meeting facet against facet, then S0025579300006082_inline1 tiles En-d in the same manner. A number of conditions, conjectured by Shephard and H. S. M. Coxeter to be equivalent to the tiling property, are also proved.

§1. Introduction. Let two probability spaces (X, , μ,) and (Y, ℬ, ν) be given. For a subset D of X × Y and a real number d ≥ 0 we consider the following problem(MP) Does there exist a measure » on X × Y having μ and ν as marginals and such that λ (D) ≥ 1 − d?This problem comes from Strassen's paper [12], where Borel probabilities on Polish spaces were treated. Further, it was investigated by many authors in more general settings (cf. [2], [4]-[7], [11]-[13]).

In 1956 Cassels proved the following result, which generalized a theorem of Marshall Hall on continued fractions. Let λ 1 …, λ r be any real numbers. Then there exists a real number α such that for all integers u > 0 and for q = 1,…,r , where C = C(r) > 0. Thus all the numbers α+ λ 1 , …, α+ λ r are badly approximable by rational numbers, which is equivalent to saying that the partial quotients in their continued fractions are bounded. In a previous paper I extended Cassels's result to simultaneous approximation. In the simplest case—that of simultaneous approximation to pairs of numbers—I proved that for any real λ 1 , …, λ r and μ 1 , …, μ r there exist α, β such that for all integers u > 0 and for q=1,…, r , where again C = C(r) > 0 . Both the construction of Cassels and my extension of it to more dimensions allow one to introduce an infinity of arbitrary choices, and consequently the set of α for (1) and the set of α, β for (2) may be made to have the cardinal of the continuum.

The purpose of this paper is to demonstrate that a number of properties of independence spaces are of finite character, thus making it possible to easily generalise known theorems for finite spaces, or matroids, to independence spaces on infinite sets.

In this note I settle a question which arose out of my first paper under the above title ( cf . [1]), where I considered the classgroup C (Z(Γ)) of the integral groupring Z(Γ) of a finite Abelian group Γ. This classgroup maps onto the classgroup C( ) of the maximal order of the rational groupring Q (Γ), and C ( ) is the product of the ideal classgroups of the algebraic number fields which occur as components of Q (Γ) and is thus in a sense known. One is then interested in the kernel D (Z(Γ)) of C (Z(Γ)) → C ( ) and in its order k (Γ). In [1] I proved that, for Γ a p -group, k (Γ) is a power of p . I also computed k (Γ) for small exponents. My computation used crucially the fact that, for the groups Γ considered, the groups of units of algebraic integers which occurred were finite, i.e. that the only number fields which turned up were Q and Q ( n ) with n 4 = 1 or n 6 = 1. The numerical results obtained led me to the question whether in fact k (Γ) tends to infinity with the order of Γ.

It has been pointed out to us by Professor L. Schoenfeld that there is a fallacy in the proof of Theorem 3 of our paper “The values of a trigonometrical polynomial at well spaced points” [ Mathematika, 13 (1966), 91–96]. The fallacy occurs in the appeal to Theorem 1 at the end of the proof. If this is to be made explicitly, we must not only put n = dn′ but also put q = dq′ but then the sum over m goes from 1 to q instead of from 1 to q′, and if one allows for this the final result becomes much weakened.

The objectjof this paper is to extend to algebraic number fields some of the recent results proved by the large sieve method. In particular we prove generalizations of Bombieri's form of the large sieve inequality [1; Theorem 1] and of the theorems of Davenport and Halberstam [2] and Bombieri [1, 4], on the average distribution of primes in arithmetic progressions.

In 1934 Gelfond [2] and Schneider [6] proved, independently, that the logarithm of an algebraic number to an algebraic base, other than 0 or 1, is either rational or transcendental and thereby solved the famous seventh problem of Hilbert. Among the many subsequent developments (cf. [4, 7, 8]), Gelfond [3] obtained, by means of a refinement of the method of proof, a positive lower bound for the absolute value of β1 log α1+β2 log α2, where β1, β2 denote algebraic numbers, not both 0, and α1,α1, denote algebraic numbers not 0 or 1, with log α1/log α2 irrational. Of particular interest is the special case in which β1,β2 denote integers. In this case it is easy to obtain a trivial positive lower bound (cf. [1; Lemma 2]), and the existence of a non-trivial bound follows from the Thue–Siegel–Roth theorem (see [4; Ch. I]). But Gelfond's result improves substantially on the former, and, unlike the latter, it is derived by an effective method of proof. Gelfond [4; p. 177] remarked that an analogous theorem for linear forms in arbitrarily many logarithms of algebraic numbers would be of great value for the solution of some apparently very difficult problems of number theory. It is the object of this paper to establish such a result.

In many investigations into the properties of convex bodies, authors have made use of distance functions ρ( K 1 K 2 ) which give a measure of the “nearness” of two convex bodies K 1 and K 2 . Sometimes they have introduced new functions to deal with particular problems. The purpose of this paper is to compare and contrast the properties of four of these functions, namely all those (so far as we are aware) which occur in the literature and have the property that they are metrics on the set of all convex bodies of some given dimension.

The object of this paper is first to generalize the basic inequality of the large sieve method to exponential sums in many variables, and then to deduce results for algebraic number fields that are analogous to known results for the rational field.