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 xx, by w∗n(x)wn∗(x) we define the supremum of all positive real values ww such that the inequality
has infinitely many solutions in algebraic real numbers aa of degree at most nn. Here H(a)H(a) means the naive height of the minimal polynomial in Z[x]Z[x] with coprime coefficients. In 1961, Wirsing asked: is it true that the quantity w∗n(x)wn∗(x) is at least n for all transcendental xx? Apart from partial results for small values of nn, this problem still remains open. Wirsing himself managed to establish the lower bound of the form w∗n(x)≥n/2+1−o(1)wn∗(x)≥n/2+1−o(1). Until recently, the only improvements to this bound were in terms of O(1)O(1). I will talk about our resent work with Schleischitz where we managed to improve the bound by a quantity of the size O(n)O(n). More precisely, we show that w∗n(x)>n/3–√wn∗(x)>n/3.
This talk is part of the online Number Theory Web Seminar, and will be streamed live on Zoom.
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Mike Bennett (University of British Columbia)
Philipp Habegger (University of Basel)
Alina Ostafe (UNSW Sydney)
Number Theory Seminar
University of Sydney
Tue, 15/09/2020 - 7:00pm
RC-4082, The Red Centre, UNSW