发表机构
Dankook University（檀国大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过构造 Engel 与 Pierce 展开数字序列间的显式同胚，建立了 Baire 范畴转移原理，并验证了 Moroz 关于映射连续性与非单调性的猜想，进而刻画了级数发散条件与收敛指数关系。
AI 中文摘要
本文构造了 Engel 展开与 Pierce 展开数字序列之间的显式对应关系，该对应关系通过将第 $n$ 个 Engel 展开数字加上 $n-2$ 得到，并证明它诱导了一个从单位区间内无理数集合到该集合的可数稠密子集的补集上的同胚。这为两种展开之间建立了 Baire 范畴的转移原理，并且我们精确确定了哪些 Borel 类被保留。随后，我们将此同胚与 Moroz (2027) 考虑的两个映射的复合等同起来，并研究第二个映射（将修正的 Engel 展开发送到 Pierce 展开）的连续性。我们确定了它的不连续点集合，证明了每个不连续点都是一个跳跃，并证明了该映射处处非单调。特别地，这证实了 Moroz (2027) 对此映射所 conjectured 的处处非单调性以及例外集之外的连续性。作为应用，我们刻画了使得相关 Pierce 展开数字级数在一个剩余集上发散的单调非增位置相关权重的特征，并关联了 Engel 与 Pierce 展开数字序列的收敛指数。
英文摘要
In this paper, we construct an explicit correspondence between Engel and Pierce expansion digit sequences, obtained by adding $n-2$ to the $n$th Engel expansion digit, and show that it induces a homeomorphism from the set of irrationals in the unit interval onto the complement of a countable dense subset of this set. This yields a transfer principle for Baire category between the two expansions, and we determine exactly which Borel classes are preserved. We then identify this homeomorphism with the composition of two maps considered by Moroz (2027), and study the continuity of the second map, which sends modified Engel expansions to Pierce expansions. We determine its set of discontinuities, show that every discontinuity is a jump, and prove that the map is nowhere monotone. In particular, this confirms, for this map, the nowhere monotonicity and the continuity outside the exceptional set conjectured by Moroz (2027). As applications, we characterize the non-increasing position-dependent weights for which the associated series of Pierce expansion digits diverges on a comeager set, and we relate the convergence exponents of the Engel and Pierce expansion digit sequences.
Comments16 pages
