Andreas Dress
Characterizing of Liouville numbers and Mahler's U-numbers in terms of Diophantine ApproximatibilityA report on joint work with Noam D. Elkies and Florian Luca
Given a real number $\alpha$, we consider the set ${\cal LM}(\alpha)$ of all positive real numbers $\tau$ for which there exist, for every $s\in\N$, infinitely many $(s+1)$-tuples $q,p_1,\dots,p_s$ of integers with $q>0$ and $|p_i - q \,\alpha^i| \le C q^{-{\frac{1}{ \tau}}}$ for some $C>0$, and show that a non-rational number $\alpha$ is algebraic of degree $m$ if and only if ${\cal LM}(\alpha)=[m,\infty)$ holds, that $\alpha$ is a Mahler $U_m$ number if and only if ${\cal LM}(\alpha)=(m,\infty)$ holds, and that ${\cal LM}(\alpha)$ is empty for all other non-rational real numbers $\alpha$. Apart from combining elementary and straightforward, yet rather lenghty computations and estimations (e.g. estimating, for a polynomial $f(x)=a_mx^m+a_{m-1}x^{m-1}+\hdots +a_0$ with inyeger coefficients, the value of $r, a_m^k\alpha ^{k+m-1}$, for any positive integer $k$, in terms of $k,r,f(\alpha), \limits ^{m}_{i=0}|a_i|$, and $\sum \limits ^{m-1}_{i=1}\lfloor r\,\alpha^i+1/2 \rfloor$) keeping good control of all the quantifyers involved , the proof makes use of some properties of an apparently new canonical $\Z$-basis of the $\Z$-module consisting of all homogeneous polynomials of degree $e$ in $m$ variables.