论文标题
Mahler功能系数的高度差距定理
A height gap theorem for coefficients of Mahler functions
论文作者
论文摘要
我们研究了具有代数系数的Mahler功率系数系数的渐近生长，该系数通过其对数Weil高度来衡量。我们表明，有五种不同的生长行为，所有这些都达到了。因此，在可能的生长中有\ emph {gaps}。在证明此高度差距定理时，我们获得了$ k $ -mahler函数为$ k $ - 当时其系数在$ o（\ log n）$中的高度时才。此外，我们推断出，在特征零的任意地面场上，$ k $ -mahler函数为$ k $ -automatic，并且仅当其系数属于有限的集合时。作为我们结果的副产品，我们还恢复了贝克尔的猜想，该猜想最近由贝尔，Chyzak，Coons和Dumas解决。
We study the asymptotic growth of coefficients of Mahler power series with algebraic coefficients, as measured by their logarithmic Weil height. We show that there are five different growth behaviors, all of which being reached. Thus, there are \emph{gaps} in the possible growths. In proving this height gap theorem, we obtain that a $k$-Mahler function is $k$-regular if and only if its coefficients have height in $O(\log n)$. Furthermore, we deduce that, over an arbitrary ground field of characteristic zero, a $k$-Mahler function is $k$-automatic if and only if its coefficients belong to a finite set. As a by-product of our results, we also recover a conjecture of Becker which was recently settled by Bell, Chyzak, Coons, and Dumas.
