Fujita’s approximation theorem in positive charactristics

A Metric Theorem for Restricted Diophantine Approximation in Positive Characteristic

0 whenever ai = 0 for all i ∈ Z, k whenever an 6= 0 and ai = 0 for i < n. We can interpret F(X) as the completion of F(X) in this absolute value. Diophantine approximation in F(X), where a generic element is approximated by elements from the field of fractions F(X), has been studied by numerous authors (the survey papers [9, 11] contain some of the known results). Broadly speaking, the object o...

.1 Kronecker's Approximation Theorem

Theorem 2 (Kronecker Approximation Theorem (1884)) ∃x ∈ Z s.t. Ax = b if and only if ∀y, y>b is an integer whenever y>A is an integral vector. Proof: To prove the forward implication, take an integral solution x∗. For any y, since y>Ax∗ = y>b, if y>A is integral then y>b must be an integer too. We next prove the converse. First, suppose that Ax = b does not have a solution (over the reals). The...

The Block Approximation Theorem

The block approximation theorem is an extensive generalization of both the well known weak approximation theorem from valuation theory and the density property of global fields in their henselizations. It guarantees the existence of rational points of smooth affine varieties that solve approximation problems of local-global type (see e.g. [HJP07]). The theorem holds for pseudo real closed field...

Analytic Approximation Theorem

An Approximation Theorem in Nonlinear Sampling

We continue here our investigation into the preservation of structural properties under the sampling of nonlinear systems. The main new result is that, under minimal hypothesis, a controllable system always satisfies a strong type of approximate sampled controllability.