A simple proof of Schmidt–Summerer’s inequality

Abstract

In this paper we give a simple proof of an inequality for intermediate Diophantine exponents obtained recently by W. M. Schmidt and L. Summerer. Introduction∞ the unit ball in sup-norm, i.e. the cube with vertices at the points (±1, . . . , ±1).W. M. Schmidt and L. Summerer [3,4] studied the asymptotic behaviour of the successive minima of the body G t B d ∞ with respect to the given lattice Λ. An appropriate choice of Λ connects this setting with the classical setting of simultaneous Diophantine approximation.In [4] Schmidt and Summerer proved important inequalities connecting the asymptotics of the first and the p-th successive minima, which lead them to an improvement of a famous Jarník's inequality between the uniform and the ordinary Diophantine exponents [2]. However, the proof they proposed was rather difficult. It uses Mahler's theory of compound bodies and involves a complicated, cumbersome analysis of special piecewise linear functions.In the present paper we give a short proof of the main result of [4]. It relies on a simple geometric observation (see Lemma 1 below) and does not use the theory of compounds.

An Optimal Bound for the Ratio Between Ordinary and Uniform Exponents of Diophantine Approximation

“…The fact that all coordinates sum up to 1 for q = 1 follows from ρ being the root of the polynomial R n,λ defined in (7). Up to change of origin and rescaling, this is the same pattern as shown by Figure 6.…”

“…Consider θ ∈ R n with Q-linearly independent coordinates with 1, and take α <λ(θ). Denote by g the unique positive root of R n,α defined by (7).…”

“…An optimal lower bound expressed as a function of the uniform exponent was established for simultaneous approximation to two real numbers and for one linear form in two variables. The question was reconsidered recently by different authors [5,7,13,17,18,26]. The optimality of Jarník's inequalities for two numbers was shown by Laurent [13].…”

“…Relation (5) is valid for the whole interval of possible values ofω(θ), but Jarník's asymptotic relation (3) is better for largeω(θ). A simple proof of (6) was given in [7].…”

Note on the spectrum of classical and uniform exponents of Diophantine approximation

“…An optimal lower bound expressed as a function of the uniform exponent was established for simultaneous approximation to two real numbers and for one linear form in two variables. The question was reconsidered recently by different authors [11,15,16,25,5,3]. The optimality of V. Jarník's inequalities for two numbers was shown by M. Laurent [11].…”