In this talk we discuss the approximation of transcendental numbers by algebraic numbers of given degree and bounded height. More precisely, for any real number x, by wn*(x) we define the supremum of all positive real values w such that the inequality
|x – a| < H(a)–w-1
has infinitely many solutions in algebraic real numbers a of degree at most n. Here H(a) means the naive height of the minimal polynomial in ℤ[x] with coprime coefficients. In 1961, Wirsing asked: is it true that the quantity wn*(x) is at least n for all transcendental x? Apart from partial results for small values of n, this problem still remains open. Wirsing himself managed to establish the lower bound of the form wn*(x) ≥ n/2+1 – o(1). Until recently, the only improvements to this bound were in terms of O(1). I will talk about our recent work with Schleischitz where we managed to improve the bound by a quantity of the size O(n). More precisely, we show that wn*(x) > n/√3.
This video is part of the Number Theory Web Seminar series.
