It has been known that mixed automorphic forms arise naturally as holomorphic forms on elliptic varieties and that they include classical automorphic forms as a special case. In this paper, we show how to construct mixed automorphic forms of type (k, l) from elliptic modular forms to give nontrivial examples of mixed automorphic forms.

In [CKR], Chan, Kim and Raghavan determine all universal positive ternary integral quadratic forms over real quadratic number fields. In this context, universal means that the form represents all totally positive elements of the ring of integers of the underlying field. This generalizes the usage of the term introduced by Dickson for the case of the ring of rational integers [D]. In the present paper, we will continue the investigation of quadratic forms with this property, considering positive quaternary forms over totally real number fields. The main goal of the paper is to prove that if E is a totally real number field of odd degree over the field of rational numbers, then there are at most finitely many inequivalent universal positive quaternary quadratic forms over the ring of integers of E. In fact, the stronger result will be proved that this finiteness holds for those forms which represent all totally positive multiples of any fixed totally positive integer. The necessity of the assumption of oddness of the degree of the extension for a general result of this type can be seen from the existence of universal ternary forms over certain real quadratic fields (for example, the sum of three squares over the field ℚ(√5), as first shown by Maass [M]).

For a fixed integer k ≥ 2 suppose we have functions Li (x) satisfying the following conditions.

Crittenden and Vanden Eynden conjectured that if n arithmetic progressions, each having modulus at least k, include all the integers from 1 to k2n-k+1, then they include all the integers. They proved this for the cases k = 1 and k = 2. We give various necessary conditions for a counterexample to the conjecture; in particular we show that if a counterexample exists for some value of k, then one exists for that k and a value of n less than an explicit function of k.

Let F(X, Y) be an absolutely irreducible polynomial with coefficients in an algebraic number field K. Denote by C the algebraic curve defined by the equation F(X, Y) = 0 and by K[C] the ring of regular functions on Cover K. Assume that there is a unit ϕ in K[C] − K such that 1 − ϕ is also a unit. Then we establish an explicit upper bound for the size of integral solutions of the equation F(X, Y) = 0, defined over K. Using this result we establish improved explicit upper bounds on the size of integral solutions to the equations defining non-singular affine curves of genus zero, with at least three points at ‘infinity’, the elliptic equations and a class of equations containing the Thue curves.

By a fundamental theorem of Brauer every irreducible character of a finite group G can be written in the field Q(εm) of mth roots of unity where m is the exponent of G. Is it always possible to find a matrix representation over its ring Z[εm] of integers? In the present paper it is shown that this holds true provided it is valid for the quasisimple groups. The reduction to such groups relies on a useful extension theorem for integral representations. Iwasawa theory on class groups of cyclotomic fields gives evidence that the answer is at least affirmative when the exponent is replaced by the order, and provides for a general qualitative result.

In this paper, we derive a number of explicit lower bounds for rational approximation to certain cubic irrationalities, proving, for example, that for any non-zero integers p and q. A number of these irrationality measures improve known results, including those for . Some Diophantine consequences are briefly discussed.

Call a set of natural numbers subset-sum-distinct (or SSD) if all pairwise distinct subsets have unequal sums. One wants to construct SSD sets in which the largest element is as small as possible. Given any SSD set, it is easy to construct an SSD set with one more element in which the biggest element is exactly double the biggest element in the original set. For any SSD set, we construct another SSD set with k more elements whose largest element is less than 2k times the largest element in the original set. This claim has been made previously for a different construction, but we show that that claim is false.

In this note we point out that a simple proof of the lower bound of the sets (b, c), and so also of Ξ(b, c), defined in the previous paper [1] can be obtained as a simple application of a general method. By Example 4.6 from [2], if [0, 1] = E0⊃E1⊃ … are sets each of which is a finite union of disjoint closed intervals such that each interval of Ek−1, contains at least mk intervals of Ek which are separated by gaps of lengths at least εk, and if mk≥2 and εk≥εk+1>0, then the dimension of the intersection of Ek is at least

Let Sk, l(Γ, ω, χ) be the space of mixed cusp forms of type (k, l) associated to a Fuchsian group Γ, a holomorphic map ω: ℋ → ℋ of the upper half plane into itself and a homomorphism χ: Γ → SL(2, R) such that ω and χ are equivariant. We construct a map from Sk, l(Γ, ω, χ) to the parabolic cohomology space of Γ with coefficients in some Γ-module and prove that this map is injective.

Let [0;a1(ξ), a2(ξ),…] denote the continued fraction expansion of ξ∈[0, 1]. The problem of estimating the fractional dimension of sets of continued fractions emerged in late twenties in papers by Jarnik [6, 7] and Besicovitch [1] and since then has been addressed by a number of authors (see [2, 4, 5, 8, 9]). In particular, Good [4] proved that the set of all ξ, for which an(ξ)→∞ as n→∞ has the Hausdorff dimension ½ For the set of continued fractions whose expansion terms tend to infinity doubly exponentially the dimension decreases even further. More precisely, let

Hirst [5] showed that dim On the other hand, Moorthy [8] showed that dim where

In this paper we investigate the solutions in integers x, y, z, X, Y, Z of the system

where P is a real number that may be taken to be arbitrarily large, and the (fixed) integer exponent h satisfies h≥4. The system has 6P3 + O(P2) “trivial” solutions in which x, y, z are a permutation of X, Y, Z. Our result implies that the number of non-trivial solutions is at most o(P3), so that the total number of solutions is asymptotic to 6P3.

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).

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.

We generalize dual reductive pairs by using reductive groups that are not necessarily subgroups of symplectic groups and construct the corresponding theta-series liftings for certain types of automorphic forms. We also discuss connections of such generalized theta-series liftings with families of abelian varieties parametrized by an arithmetic variety.

An inhomogeneous version of a general form of the Jarník-Besicovitch Theorem is proved.

In 1984 Heath-Brown [5] proved the following conjecture of Erdős and Mirsky [2] (which seemed at one time as hard as the twin prime problem):

“There exist infinitely many integers n for which d(n) = d(n + 1).”

On Waring's problem for cubes, it is conjectured that every sufficiently large natural number can be represented as a sum of four cubes of natural numbers. Denoting by E(N) the number of the natural numbers up to N that cannot be written as a sum of four cubes, we may express the conjecture as E(N)≪1.

The shape of large densest sphere packings in a lattice L ⊂ Ed (d ≥ 2), measured by parametric density, tends asymptotically not to a sphere but to a polytope, the Wulff-shape, which depends only on L and the parameter. This is proved via the density deviation, derived from parametric density and diophantine approximation. In crystallography the Wulff-shape describes the shape of ideal crystals. So the result further indicates that the shape of ideal crystals can be described by dense lattice packings of spheres in E3.

In 1973, Montgomery [12] introduced, in order to study the vertical distribution of the zeros of the Riemann zeta function, the pair correlation function

where w(u) = 4/(4 + u2) and γjj = 1, 2, run over the imaginary part of the nontrivial zeros of ζ(s). It is easy to see that, for T → ∞,

uniformly in X, and Montgomery [12], see also Goldston-Montgomery [7], proved that under the Riemann Hypothesis (RH)

uniformly for X ≤ T ≤ XA, for any fixed A > 1. He also conjectured, under RH, that (1) holds uniformly for Xε ≤ T ≤ X, for every fixed ε > 0. We denote by MC the above conjecture.