A new exponent of simultaneous rational approximation

Abstract

We introduce a new exponent of simultaneous rational approximation λmin(ξ, η) for pairs of real numbers ξ, η, in complement to the classical exponents λ(ξ, η) of best approximation, and λ(ξ, η) of uniform approximation. It generalizes Fischler's exponent β0(ξ) in the sense that λmin(ξ, ξ 2 ) = 1/β0(ξ) whenever λ(ξ, ξ 2 ) = 1. Using parametric geometry of numbers, we provide a complete description of the set of values taken by (λ, λmin) at pairs (ξ, η) with 1, ξ, η linearly independent over Q.

Exponents of diophantine approximation in dimension 2 for numbers of Sturmian type

“…See [11] for the motivation of the division by H(α) in the left-hand side. We now recall the definitions of the last two exponents β 0 (ξ) and λ min (ξ), introduced respectively by Fischler in [19] and by the author in [24] on the basis of [19]. Set Ξ = (1, ξ, ξ 2 ) and for each 0 ≤ µ < λ(ξ), denote by λ µ (Ξ) the supremum of the real numbers λ for which…”

“…In particular λ min (ξ) ≤ λ 2 (ξ). Note that the definition of λ min in [24] applies to general points Ξ = (1, ξ, η) ∈ R 3 . In the current situation, the two above exponents are connected in the following way: if β 0 (ξ) < 2, then λ 2 (ξ) = 1 and λ min (ξ) = 1/β 0 (ξ) (see [24,Lemma 1.3]).…”

“…Note that the definition of λ min in [24] applies to general points Ξ = (1, ξ, η) ∈ R 3 . In the current situation, the two above exponents are connected in the following way: if β 0 (ξ) < 2, then λ 2 (ξ) = 1 and λ min (ξ) = 1/β 0 (ξ) (see [24,Lemma 1.3]). The classical general estimates below are valid for each ξ ∈ R which is neither rational nor quadratic.…”

“…By the above, the sequence of primitive points y ∈ Z 3 such that y ∧ Ξ ≤ y −µ coincides, up to a finite numbers of terms, with the sequence (y i ) i≥0 . We deduce by a classical reasoning (see for example [24,Section 2]) that…”

“…The exponent λ min (ξ) above is defined in [24] and related to work of Fischler [19]. We recall its definition in the next section.…”