An equivalence principle between polynomial and simultaneous Diophantine approximation

Abstract

We show that Mahler's classification of real numbers ζ with respect to the growth of the sequence (w n (ζ)) n≥1 is equivalently induced by certain natural assumptions on the decay of the sequence (λ n (ζ)) n≥1 concerning simultaneous rational approximation. Thereby we obtain a much clearer picture on simultaneous approximation to successive powers of a real number in general. Another variant of the Mahler classification concerning uniform approximation by algebraic numbers is derived as well. Our method has several applications to classic exponents of Diophantine approximation and metric theory. We deduce estimates on the Hausdorff dimension of well-approximable vectors on the Veronese curve and refine the best known upper bound for the exponent λ n (ζ) for even n ≥ 4.

Cubic approximation to Sturmian continued fractions

“…In contrast to Theorem 2.1 and Theorem 2.2, which are generalizations of [19, Theorem 2.1 and Theorem 2.2] with similar proofs, no variant of the next theorem was mentioned in [19]. It is essentially derived from specializing recent results from [22] and [25]. Theorem 2.3.…”

“…The bounds are smaller than λ 3 (ζ) = 1/ √ 5 ≈ 0.4472, however considerably larger than the expected value ( √ 5 − 1)/4 ≈ 0.3090 for λ 4 (ζ), see [19,Theorem 2.3] and the enclosed comments. Concerning (23), we refer to [25,Section 4.3] for a semi-effective minimum rate at which the limit 0 is approached, and analogous results for more general classes of numbers.…”

“…If we weaken the assumptions by only prescribing some upper bound less than 2n − 1 for w n−1 (ζ) and/or replacing the right hand side in (25) by a larger number, we still obtain upper bounds for w 2n−1 (ζ) and λ 2n−1 (ζ). Moreover, upon the assumptions of the theorem the condition (25) in fact implies that the left hand side of (25) must tend to 1 as i → ∞. We could equivalently impose |nν −1 (25).…”

“…Moreover, upon the assumptions of the theorem the condition (25) in fact implies that the left hand side of (25) must tend to 1 as i → ∞. We could equivalently impose |nν −1 (25). Furthermore the left condition in (24) is in fact redundant as it can be derived from the right condition in (24) combined with (25).…”

Diophantine Approximation in Prescribed Degree

“…In fact the semi-effective exponential upper bounds for the growth of the sequence (w n (ζ)) n≥1 from [1, Théorème 4.2] apply to any ζ which satisfies w 2 (ζ) > 2. See also [40,Corollary 4.6] for a stronger upper bound for the exponent w 3 (ζ) of the form w 3 (ζ) < 15/( w 2 (ζ) − 2) 2 as soon as w 2 (ζ) > 2. We will discuss generalizations of these phenomena in Problem 3 in Section 6 below.…”