Independence Inheritance and Diophantine Approximation for Systems of Linear Forms

Metric Diophantine Approximation for Systems of Linear Forms via Dynamics

The goal of this paper is to generalize the main results of [KM1] and subsequent papers on metric Diophantine approximation with dependent quantities to the set-up of systems of linear forms. In particular, we establish ‘joint strong extremality’ of arbitrary finite collection of smooth nondegenerate submanifolds of R. The proofs are based on quantitative nondivergence estimates for quasi-polyn...

Diophantine Approximation of Ternary Linear Forms . II

Let 6 denote the positive root of the equation xs + x2 — 2x — 1 = 0; that is, 8 = 2 cos(27r/7). The main result of the paper is the evaluation of the constant lim supm-co min M2\x + By + 02z|, where the min is taken over all integers x, y, z satisfying 1 g max (\y\, |z|) g M. Its value is (29 + 3),/7 = .78485. The same method can be applied to other constants of the same type.

Inhomogeneous Non-linear Diophantine Approximation

Let be a strictly positive monotonically decreasing function deened on the set of positive integers. Given real numbers and , consider the solubility of the following two inequalities jq + pj < (q); (1) jq + p + j < (q) (2) for integers p and q. The rst problem is said to be homogeneous and the second inho-mogeneous (see 2]). The well known theorem of Khintchine 2, 4] asserts that for almost al...

On Solving Linear Diophantine Systems Using Generalized Rosser's Algorithm

On the Approximation of Pseudo Linear Systems by Linear Time Varying Systems (RESEARCH NOTE)

This paper presents a modified method for approximating nonlinear systems by a sequence of linear time varying systems. The convergence proof is outlined and the potential of this methodology is discussed. Simulation results are used to show the effectiveness of the proposed method.