In this paper we investigate the metrical structure of the set of all points $X\in\mathbb{R}^n$ which satisfy a simultaneously small system of Diophantine inequalities for infinitely many integer vectors. We establish the complete metric theory for the given system which implies a general Khintchine--Groshev type theorem, as well as its Hausdorff measure generalization. The latter includes the original dimension results obtained in [5] as special cases.

Let $k$ be a number field and $Cl(k)$ its class group. Let $\Gamma$ be a non trivial finite $2$-group. Let $R_m(k, \Gamma)$ be the subset of $Cl(k)$ consisting of those classes which are realizable as Steinitz classes of tame Galois extensions of $k$ with Galois group isomorphic to $\Gamma$. In the present article, we show that $R_m(k,\Gamma)$ is the full group $Cl(k)$, if the class number of $k$ is odd. We study an embedding problem connected with Steinitz classes in the perspective of studying realizable Galois module classes, when $\Gamma$ is defined by certain central non-split group extensions, examples of which are certain groups of order $32$ or $64$. For such groups $\Gamma$, We prove that for all $c\in Cl(k)$, there exist a tame quadratic extension of $k$, with Steinitz class $c$, and which is embeddable in a~Galois extension of $k$ with Galois group isomorphic to $\Gamma$.

Let \[ f(x_1,...,x_n)=a_{1}x_{1}^{k}+\cdots+a_{n}x_{n}^{k}\\ g(x_1,...,x_n)=b_{1}x_{1}^{k}+\cdots+b_{n}x_{n}^{k} \] be a pair of additive forms of degree $k=p^{\tau}(p-1)$. We are interested in finding conditions which guarantee the existence of $p$-adic zeros for this pair of forms. A well-known conjecture due to Emil Artin states that the condition $n > 2k^2$ is sufficient. Here we prove that $$n > 2\left(\frac{p}{p-1}\right)k^2-2k$$ is sufficient, provided that $p > 5$ and $\tau \geq \dfrac{p-1}{2}$.

We prove new bounds for weighted mean values of sums involving Fourier coefficients of cusp forms that are automorphic with respect to a~Hecke congruence subgroup $\Gamma\leq SL(2,{\mathbb Z}[i])$, and correspond to exceptional eigenvalues of the Laplace operator on the space $L^2(\Gamma\backslash SL(2,{\mathbb C})/SU(2))$. These results are, for certain applications, an effective substitute for the generalised Selberg eigenvalue conjecture. We give a~proof of one such application, which is an upper bound for a~sum of generalised Kloosterman sums (of significance in the study of certain mean values of Hecke zeta-functions with groessencharakters). Our proofs make extensive use of Lokvenec-Guleska's generalisation of the Bruggeman-Motohashi summation formulae for $PSL(2,{\mathbb Z}[i])\backslash PSL(2,{\mathbb C})$. We also employ a~bound of Kim and Shahidi for the first eigenvalues of the relevant Laplace operators, and an `unweighted' spectral large sieve inequality (our proof of which is to appear separately).