Seminar in Number Theory

主講者: 陳慧錚博士 (中研院數學所)
講題: Diophantine approximation and Hausdorff dimension in positive characteristic
時間: 2012-09-18 (Tue.)  15:00 - 16:00
地點: 數學所 722 研討室 (台大院區)
Abstract: For $\alpha \in \mathbb{R}$, let $||\alpha||$ denotes the distance of $\alpha$ from the nearest integer. Mahler (1932) conjectured that almost all real numbers $\alpha$ has the following property (equivalent to Mahler's original conjecture by a classical transference principle ): for any positive integer $n$ and any $\epsilon >0$ there exist only finitely many integral polynomials $P(x)$ with degree $n$ such that
$ \left | P(\alpha ) \right |< H^{-n-\epsilon } $
where $H$ denotes the height of $P.$ Sprind$\breve{z}$uk (1962) finally succeeded in establishing Mahler's conjecture. Later (1966) Baker gave a more precise results different from $H^{-n-\epsilon }$ using a modified version of Sprind$\breve{z}$uk's method. In the first talk I will give an analogue of Baker's theorem in positive characteristic. In the next talk we will investigate the sets related to the approximation by algebraic elements of bounded degrees. Baker-Schmidt's theorem derive the Hausdorff dimension of these sets. We will prove Baker-Schmidt's theorem for function field version via the three crucial Lemmas: Baker's theorem, Minkowski's geometry numbers and reguler system.
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