Compound Multiplicative Diophantine Systems

Multiplicative Diophantine approximation

In his paper, Dirichlet gives a complete proof for n = 1 and observes that this proof can be easily extended to arbitrary values of n. Good references on this topic are Chapter II of [52] and Cassels’ book [17]. There are in the literature many papers on various generalisations of the Dirichlet Theorem and on closely related problems. A typical question asks whether for a given set of mn real n...

Multiplicative Diophantine Exponents of Hyperplanes

We study multiplicative Diophantine approximation property of vectors and compute Diophantine exponents of hyperplanes via dynamics.

Asymptotic Diophantine Approximation: the Multiplicative Case

Let α and β be distinct badly approximable numbers, and let ψ : [1,∞)→ [1,∞) be a monotone increasing unbounded function. We prove an asymptotic formula for the number of positive integers q < Q that satisfy ‖qα‖‖qβ‖ ≤ ψ(Q)/Q.

Multiplicative Diophantine Exponents of Hyperplanes and their Nondegenerate Submanifolds

We consider multiparameter dynamics on the space of unimolular lattices. Along with quantitative nondivergence we prove that multiplicative Diophantine exponents of hyperplanes are inherited by their nondegenerate submanifolds.

On Solving Linear Diophantine Systems Using Generalized Rosser's Algorithm