Rational approximation to real points on quadratic hypersurfaces

Abstract

Let Z be a quadratic hypersurface of double-struckPnfalse(double-struckRfalse) defined over double-struckQ containing points whose coordinates are linearly independent over double-struckQ. We show that, among these points, the largest exponent of uniform rational approximation is the inverse 1/ρ of an explicit Pisot number ρ<2 depending only on n if the Witt index (over double-struckQ) of the quadratic form q defining Z is at most 1, and that it is equal to 1 otherwise. Furthermore, there are points of Z which realize this maximum. They constitute a countably infinite set in the first case, and an uncountable set in the second case. The proof for the upper bound 1/ρ uses a recent transference inequality of Marnat and Moshchevitin. In the case n=2, we recover results of the second author while for n>2, this completes recent work of Kleinbock and Moshchevitin.

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

“…Recall that the sequence s is not necessarily bounded and that U denotes the map introduced in Section 3.3. We define the sequence (p k ) k≥−1 by (25) (p −1 , p 0 ) = (0, 1) and p k+1 = s k+1 p k + p k−1 (k ≥ 0). Proposition 4.6.…”

“…The proof of this result is at the end of this section. With that goal in mind, let us introduce for each i ∈ N the sequence (q (25). Moreover, the theory of continued fractions (see for instance [32, Chapter I]) ensures that for each k ≥ i > 0 we have (29) q…”

“…The main result of this section, namely Theorem 4.11 below, will help dealing with this problem. Recall that (p k ) k≥−1 is defined by (25) and U is the morphism introduced in Section 3.3. Theorem 4.11.…”

“…The set of points (1, η, η 2 ) with η ∈ R corresponds to a quadratic hypersurface associated to the quadratic form x 0 x 2 − x 2 1 . As a consequence of [25] (see [25,Theorem 7.3]), for each η with 0 < η < 1, there is ε ′ > 0 with the following property. Let ξ ∈ R (which is neither rational nor quadratic) with λ 2 (ξ) ≥ 1/γ −ε ′ , and write Ξ = (1, ξ, ξ 2 ).…”

Quantitative rational approximation on spheres

“…This subject has some vintage, having been considered by Lang [Lan65], and has seen considerable activity recently. We refer the reader to [GGN13, GGN14, GK17] and [GGN20] for results on metric Diophantine approximation on homogeneous varieties of semisimple groups, to [KM15, KdS18, KM19, Mos16, Sar19, SW19] for results on spheres, and to [FKMS14,PR19] for results on quadratic surfaces. Intrinsic Diophantine approximation on spheres has received particular attention, both for its own sake and for connections to quantum gates as pointed out by Sarnak [Sar15] and to computer science [BS17].…”