A note on Engel series expansions of Laurent series and Hausdorff dimensions

作者：Lu, Meiying^*

来源：Journal of Number Theory, 2014, 142: 44-50.

DOI：10.1016/j.jnt.2014.02.014

• Engel series expansions
• Formal Laurent series
• Hausdorff dimension

Let F-q be the finite field with q elements and F-q((z(-1))) be the field of all formal Laurent series with coefficients in F-q. For any x is an element of I := z(-1)F(q)((z(-1))), the Engel series expansion of x is Sigma(infinity)(n=1) 1/a(1)(x)...a(n)(x) with a(j)(x) is an element of F-q[z]. Suppose that phi : N -> R+ is a function satisfying phi(n) >= n for all integers n large enough. In this note, we consider the following set
E(phi) = {x is an element of I: lim(n ->infinity) deg a(n)(x)/phi(n) = 1},
and establish a lower bound of its Hausdorff dimension. As a direct application, we obtain in particular dim(II) {x is an element of I: lim(n ->infinity) deg a(n)(x)/n(beta) = gamma} = 1 (where beta > 1, gamma > 0 or beta = 1, gamma >= 1, and dimu denotes the Hausdorff dimension), which generalizes a result of J. Wu dated 2003.

• 出版日期2014-9
• 单位重庆师范大学