Simultaneous Approximation to Two Reals: Bounds for the Second Successive Minimum

Abstract

Abstract. Introduced in Schmidt and Summerer [Parametric geometry of numbers and applications. Acta Arith. 140 (2009), 67-91], approximation exponents ϕ i , ϕ i , (i = 1, 2, 3), attached to points ξ = (ξ 1 , ξ 2 ) in R 2 , give information on Diophantine approximation properties of these points. Laurent [Exponents of Diophantine approximation in dimension two. Canad. J. Math. 61 (2009), 165-189] had described all possible quadruples (ϕ 1 , ϕ 1 , ϕ 3 , ϕ 3 ) arising in this way. Our emphasis here will be on ϕ 2 , ϕ 2 and the construction of suitable "3-systems".

On a question of Schmidt and Summerer concerning 3-systems

“…Alternatively to the above examples, the pure existence of pairs (ξ 1 , ξ 2 ) inducing ϕ 3 = 1 (or ϕ 3 = 1) also follows from Roy's results [2] and [3,Theorem 11.5] (the latter result, already quoted in [9], provides an explicit description of the spectrum of sixtuples ϕ 1 , ϕ 2 , ϕ 3 , ϕ 1 , ϕ 2 , ϕ 3 by a system of complicated inequalities). The main concern of the question of Schmidt and Summerer appears to be the construction of a suitable 3-system, carried out in Section 2.1 below.…”

“…Schmidt and Summerer enclose a remark to Theorem 1 pointing out that in the case ϕ 3 = 1 excluded in its claim, we have ϕ 2 = 1 and ϕ 1 = ϕ 2 = −1/2 (by mistake they denoted −1/3 instead in [9]). However, there is a gap in Theorem 1 concerning the existence of graphs with the property ϕ 3 = 1, and related real numbers ξ 1 , ξ 2 .…”

“…Schmidt and Summerer [9] recently provided additional information by including the second successive minimum in the picture.…”

“…However, there is a gap in Theorem 1 concerning the existence of graphs with the property ϕ 3 = 1, and related real numbers ξ 1 , ξ 2 . In [9] they state "But one really should prove that ξ = (ξ 1 , ξ 2 ) with (1, ξ 1 , ξ 2 ) linearly independent over Q with ϕ 3 = 1 exist. We invite the reader to construct a proper 3-system P with this property."…”

“…While the results in [4] are originally formulated in the language of another type of exponents, the two types of exponents determine each other via the identities of [6, Theorem 1.4], and we derive Theorem 2. We note that for the sole purpose of ϕ 3 = 1, as desired in [9] and rephrased above, in fact any numbers ξ 1 , ξ 2 which are simultaneously approximable to any order by rational numbers can be chosen. In particular, one may choose the pair (ξ, ξ 2 ) with ξ any Liouville number, see [5,Theorem 3.1].…”