Engel Series Expansions of Laurent Series and Hausdorff Dimensions
Engel Series Expansions of Laurent Series and Hausdorff Dimensions
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DOI：
10.1017/s1446788700003438
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发表时间：
2003-08
影响因子：
0.7
通讯作者：
Jun Wu
中科院分区：
文献类型：
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作者：
Jun Wu
摘要对于任意正整数q≧2，设Fq是一个有q个元素的有限域，Fq （(z-1)）是一个不定z中的所有形式洛朗级数的域，我表示形式幂级数Fq （(z-1)）经P(l) = 1归一化的环中的值理想z- 1fq [[z-1]]。对于任意x∈I，设该级数为x的Laurent级数的恩格尔展开式。Grabner和Knopfmacher证明了集合A(α) = {x∞I: limn→∞deg and (x)/n = α}当α = q/(q -l)时p测度为l，其中deg and (x)是多项式an(x)的次幂。在本文中，我们证明了对于任意α≧l， A（α）具有Hausdorff维数l。除此之外，我们还证明了对于任意整数m，下面的集合B(m) = {x∈l: deg and +1(x) - deg and (x) = m对于任意n≧l}具有Hausdorff维数1。
Abstract For any positive integer q≧2, let Fq be a finite field with q elements, Fq ((z-1)) be the field of all formal Laurent series in an inderminate z, I denote the valuation ideal z-1Fq [[z-1]] in the ring of formal power series Fq ((z-1)) normalized by P(l) = 1. For any x ∈ I, let the series be the Engel expansin of Laurent series of x. Grabner and Knopfmacher have shown that the P-measure of the set A(α) = {x ∞ I: limn→∞ deg an(x)/n = ά} is l when α = q/(q -l), where deg an(x) is the degree of polynomial an(x). In this paper, we prove that for any α ≧ l, A(α) has Hausdorff dimension l. Among other thing we also show that for any integer m, the following set B(m) = {x ∈ l: deg an+1(x) - deg an(x) = m for any n ≧ l} has Hausdorff dimension 1.