Diophantine Approximation, Fractal Geometry and Related topics / Approximation diophantienne, géométrie fractale et sujets connexes

Damien Roy

4 juin 2024, 14:10

1h

Auditorium Maurice GROSS (Marne)

Description

Parametric geometry of numbers and simultaneous approximation to geometric
progressions

An important problem in Diophantine approximation is to determine, for
a given positive integer n, the supremum λ􏰖n of the exponents λ􏰖n(ξ) of uniform simultaneous rational approximation to geometric progressions (1, ξ, ξ2, . . . , ξn) whose ratio ξ is either a transcendental real number or an algebraic real number of degree > n. In 1969, Davenport and Schmidt provided an upper bound on λ􏰖n and, via geometry of numbers, they deduced a corresponding lower bound on the exponent of best approximation to such ξ by algebraic integers of degree at most n + 1. The same general transference principle applies to other classes of numbers, like approximation to ξ by algebraic units of degree at most n + 2, as Teuli ́e showed in 2001. Recall that Dirichlet’s theorem on simultaneous rational approximation yields λ􏰖n ≥ 1/n. However, we still don’t know, for any n ≥ 3, if λ􏰖n is equal to 1/n or strictly greater.
Inthistalk,weconcentrateonthecasesn=2andn=3. Forn=2,Ishowedin 2003 that the upper bound of Davenport and Schmidt for λ􏰖2 is best possible, namely that λ􏰖2 = 1/γ ∼= 0.618, where γ stands for the golden ratio. Then, for many years, I thought that λ􏰖3 could be equal to the positive root λ3 ∼= 0.4245 of the polynomial T 2 − γ3T + γ, until I realized that it is strictly smaller. As the argument lead only to a very small improvement on the upper bound, I simply published, in 2008, the proof that λ􏰖3 ≤ λ3.
In the presentation, we take the point of view of parametric geometry of numbers. We first recall the basic facts that we need about n-systems and dual n-systems. For n = 2, we explain why a point (1,ξ,ξ2) with optimal exponent λ􏰖2(ξ) = 1/γ admits a very simple self-similar dual 3-system, we give generic algebraic relations between the points of Z3 that realize this map up to a bounded difference, and we show how these in turn determine the point (1, ξ, ξ2). One can hope that a similar phenomenon holds for each n ≥ 2. For n = 3, assuming that λ􏰖3(ξ) = λ3, we find an interesting self-similar dual 4-system attached to the point (1,ξ,ξ2,ξ3) and algebraic relations with similar properties between the points that realize it up to bounded difference. However, they eventually lead to a contradiction. . .
In general, the theory attaches a dual n-system P = (P1,...,Pn): [0,∞) → Rn to
any non-zero point u of Rn, and P is unique up to bounded difference. This encodes
most of the Diophantine approximation properties of u. For a geometric progression
u = (1,ξ,ξ2,ξ3) in R4 with λ􏰖3(ξ) >
√
2 − 1 ∼= 0.4142, we can show that the behavior of P is qualitatively much simpler than that of a general dual 4-system. Moreover, the differences P3(q) − P1(q) and P4(q) − P2(q) both tend to infinity with q. Based on this, we deduce the existence of a sequence of integral bases of R4 which, in a simple way, realize P up to a bounded difference. We propose this as a tool to improve the present upper bound λ3 on λ􏰖3(ξ). By contrast, the current way of studying λ􏰖n(ξ) for a general n is to form a sequence of so-called minimal points for u = (1,ξ,...,ξn), which can be loosely described as a sequence of points of Zn+1 that realize the first component P1 of P up to bounded difference.