On Exponents of Homogeneous and Inhomogeneous Diophantine Approximation
نویسنده
چکیده
In Diophantine Approximation, inhomogeneous problems are linked with homogeneous ones by means of the so-called Transference Theorems. We revisit this classical topic by introducing new exponents of Diophantine approximation. We prove that the inhomogeneous exponent of approximation to a generic point in R by a system of n linear forms is equal to the inverse of the uniform homogeneous exponent associated to the system of dual linear forms. 2000 Math. Subj. Class. 11J20, 11J13, 11J82.
منابع مشابه
Exponents of Inhomogeneous Diophantine Approximation
– In Diophantine Approximation, inhomogeneous problems are linked with homogeneous ones by means of the so-called Transference Theorems. We revisit this classical topic by introducing new exponents of Diophantine approximation. We prove that the exponent of approximation to a generic point in R n by a system of n linear forms is equal to the inverse of the uniform homogeneous exponent associate...
متن کاملInhomogeneous theory of dual Diophantine approximation on manifolds Dedicated to Bob Vaughan on his 65th birthday
The theory of inhomogeneous Diophantine approximation on manifolds is developed. In particular, the notion of nice manifolds is introduced and the divergence part of the Groshev type theory is established for all such manifolds. Our results naturally incorporate and generalize the homogeneous measure and dimension theorems for non-degenerate manifolds established to date. The results have natur...
متن کاملA Note on Metric Inhomogeneous Diophantine Approximation
An inhomogeneous version of a general form of the Jarn k-Besicovitch Theorem is proved. Dedicated to Professor F. Chong for his 80th birthday 1. Introduction In some respects, inhomogeneous Diophantine approximation is rather diierent from homogeneous Diophantine approximation. Results in the former, where the additional variables ooer extràdegrees of freedom', are sometimes sharper or easier t...
متن کاملDiophantine Exponents of Affine Subspaces: the Simultaneous Approximation Case
We apply nondivergence estimates for flows on homogeneous spaces to compute Diophantine exponents of affine subspaces of Rn and their nondegenerate submanifolds.
متن کاملSimultaneous inhomogeneous Diophantine approximation on manifolds
In 1998, Kleinbock & Margulis [KM98] established a conjecture of V.G. Sprindzuk in metrical Diophantine approximation (and indeed the stronger Baker-Sprindzuk conjecture). In essence the conjecture stated that the simultaneous homogeneous Diophantine exponent w0(x) = 1/n for almost every point x on a non-degenerate submanifold M of Rn. In this paper the simultaneous inhomogeneous analogue of Sp...
متن کامل