Full Dimensional Sets of Reals Whose Sums of Partial Quotients Increase in Certain Speed

For a real x∈(0,1)∖Q, let x=[a1(x),a2(x),⋯] be its continued fraction expansion. Let sn(x)=∑j=1naj(x). The Hausdorff dimensions of the level setsEφ(n),α:={x∈(0,1):limn→∞⁡sn(x)φ(n)=α} for α≥0 and a non-decreasing sequence {φ(n)}n=1∞ have been studied by E. Cesaratto, B. Vallée, J. Wu, J. Xu, G. Iommi, T. Jordan, L. Liao, M. Rams et al. In this work we carry out a kind of inverse project of their work, that is, we consider the conditions on φ(n) under which one can expect a 1-dimensional set Eφ(n),α. We give certain upper and lower bounds on the increasing speed of φ(n) when Eφ(n),α is of Hausdorff dimension 1 and a new class of sequences {φ(n)}n=1∞ such that Eφ(n),α is of full dimension. For an irregular sequence {φ(n)}n=1∞, a full dimensional set Eφ(n),α is impossible.