Multiplicative Diophantine approximation
نویسنده
چکیده
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 numbers αij , 1 ≤ i ≤ n, 1 ≤ j ≤ m, the above statement continues to hold with an exponent of Q in (1.2) smaller than −m/n. In most of the works, the sup norm is used, exactly as in (1.1) and (1.2). However, it follows from (1.1) and (1.2) that
منابع مشابه
Multiplicative Diophantine Exponents of Hyperplanes
We study multiplicative Diophantine approximation property of vectors and compute Diophantine exponents of hyperplanes via dynamics.
متن کاملDiophantine Approximation with Arithmetic Functions, I
We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.
متن کاملAsymptotic 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.
متن کاملExpanding Translates of Curves and Dirichlet-minkowski Theorem on Linear Forms
We show that a multiplicative form of Dirichlet’s theorem on simultaneous Diophantine approximation as formulated by Minkowski, cannot be improved for almost all points on any analytic curve on R which is not contained in a proper affine subspace. Such an investigation was initiated by Davenport and Schmidt in the late sixties. The Diophantine problem is then settled via showing that certain se...
متن کاملMultiplicative 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.
متن کاملSelf-similar fractals and arithmetic dynamics
The concept of self-similarity on subsets of algebraic varieties is defined by considering algebraic endomorphisms of the variety as `similarity' maps. Self-similar fractals are subsets of algebraic varieties which can be written as a finite and disjoint union of `similar' copies. Fractals provide a framework in which, one can unite some results and conjectures in Diophantine g...
متن کامل