In 1970 A. Baker and W. Schmidt introduced the concept of a regular system of numbers and vectors. They proved that the set of real algebraic numbers forms a regular system on any fixed interval. This fact has allowed to prove some important results in the metric theory of transcendental numbers. In this paper their approach is applied to the set of algebraic integers α in short intervals of le...

The method of constructing the generalized dihedral group as a semidirect product an abelian and Z2 integers modulo 2 is extended to case gyrogroups. This leads study new class gyrogroups, which includes groups special case. In this article, we show that any dihedralizable gyrogroup can be enlarged dihedralized gyrogroup. Then, establish algebraic properties gyrogroups well combinatorial their ...

Let n be an integer ≥1 and let θ be a real number which is not an algebraic number of degree ≤ n/2 . We show that there exist >0 and arbitrary large real numbers X such that the system of linear inequalities |x0|≤X and |x0θ−xj |≤ X−1/ n/2 for 1≤j≤n, has only the zero solution in rational integers x0,...,xn. This result refines a similar statement due to H. Davenport and W. M. Schmidt, where the...

Many cryptographic schemes have been established based on the hardness of lattice problems. For the asymptotic efficiency, ideal lattices in the ring of cyclotomic integers are suggested to be used in most such schemes. On the other hand in computational algebraic number theory one of the main problem is called principle ideal problem (PIP). Its goal is to find a generators of any principle ide...

The ring Z consists of the integers of the field Q, and Dedekind takes the theory of unique factorization in Z to be clear and well understood. The problem is that unique factorization can fail when one considers the integers in a finite extension of the rationals, Q(α). Kummer showed that when Q(α) is a cyclotomic extension (i.e. α is a primitive pth root of unity for a prime number p), one ca...