Diophantine Approximation in Prescribed Degree

Abstract

We investigate approximation to a given real number by algebraic numbers and algebraic integers of prescribed degree. We deal with both best and uniform approximation, and highlight the similarities and differences compared with the intensely studied problem of approximation by algebraic numbers (and integers) of bounded degree. We establish the answer to a question of Bugeaud concerning approximation to transcendental real numbers by quadratic irrational numbers, and thereby we refine a result of Davenport and Schmidt from 1967. We also obtain several new characterizations of Liouville numbers, and certain new insights on inhomogeneous Diophantine approximation. As an auxiliary side result, we provide an upper bound for the number of certain linear combinations of two given relatively prime integer polynomials with a linear factor. We conclude with several open problems.

On Wirsing's problem in small exact degree

“…We return to approximation in exact degree, especially our problem if w * =n (ξ) ≥ n holds for any transcendental real ξ. For n = 2, it was shown in [26], refining a result of Moshchevitin [18] (which in turn refined on Jarník [16]) to exact degree, that even the stronger estimate (5) w * =2 (ξ) ≥ w 2 (ξ) • ( w 2 (ξ) − 1) ≥ 2 holds. This is sharp in the non-trivial case when ξ is a so-called extremal number [20].…”

“…Theorem 1.3 was known for n = 2 in view of ( 5) and (9). The claim for n = 3 also occurs in [26]. However, there was a mistake in the latter proof.…”

“…For any n > 2 the problem is open (as is Wirsing's problem). Contributions on this subject have so far been obtained by Bugeaud and Teulie [9], see also [26], [30]. Building up on ideas by Davenport and Schmidt [12] it is implicitly shown in [9] that (6) w * =n (ξ) ≥…”

“…Similar to (4), the standard argument Proposition 2.7 yields for every ξ the estimate (9) w =n (ξ) ≥ w * =n (ξ). As noticed in [26], the exponent w =n (ξ) would coincide with w n (ξ) if we omit the irreduciblity condition on P in Definition 3, by multiplying polynomials derived from Dirichlet's Theorem with suitable powers of the variable T if needed. The same holds when we do not restrict to exact degree, see Lemma 2.4 below.…”

An equivalence principle between polynomial and simultaneous Diophantine approximation

“…For example one can fix the degree of the algebraic numbers in (15) equal to n, or restrict to approximation by algebraic integers or algebraic units. See for example [16], [17], or [33].…”