Abstract
We provide a lower bound for the ratio between the ordinary and uniform exponent of both simultaneous Diophantine approximation and Diophantine approximation by linear forms in any dimension. This lower bound was conjectured by Schmidt and Summerer and already shown in dimension $2$ and $3$. This lower bound is reached at regular graph presented in the context of parametric geometry of numbers, and thus optimal.
Highlights
In dimension n = 1, Khintchine [8] observed that the uniform exponent always takes the value 1 and it follows from Dirichlet’s Schubfachprinzip that the ordinary exponent satisfy ω(θ) = λ(θ) ≥ 1 = ω(θ) = λ(θ)
Schmidt and Summerer provided an alternative proof using parametric geometry of numbers in [20], and the following bound for approximation by linear forms: ω(θ) 4ω(θ) − 3 − 1
These bounds apply in a more general setting of simultaneous Diophantine approximation by a set of linear forms. Using their new tools of parametric geometry of numbers, Schmidt and Summerer [18] provided the first general improvement working in the whole admissible interval of values of the uniform exponents ωand λ
Summary
Schmidt and Summerer provided an alternative proof using parametric geometry of numbers in [20], and the following bound for approximation by linear forms: ω(θ) 4ω(θ) − 3 − 1. These bounds apply in a more general setting of simultaneous Diophantine approximation by a set of linear forms Using their new tools of parametric geometry of numbers, Schmidt and Summerer [18] provided the first general improvement working in the whole admissible interval of values of the uniform exponents ωand λ. Θn are Q-linearly independent and λ(θ) = λand λ(θ) = Cλ It follows from Roy’s theorem [16] applied to Schmidt-Summerer’s regular graphs [20] [15] that the lower bound is reached and optimal.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
