Full dimensional sets of reals whose sums of partial quotients increase in certain speed

Abstract For a real x ∈ ( 0 , 1 ) ∖ Q , let x = [ a 1 ( x ) , a 2 ( x ) , ⋯ ] be its continued fraction expansion. Let s n ( x ) = ∑ j = 1 n a j ( x ) . The Hausdorff dimensions of the level sets E φ ( n ) , α : = { x ∈ ( 0 , 1 ) : lim n → ∞ ⁡ s n ( x ) φ ( n ) = α } for α ≥ 0 and a non-decreasing sequence { φ ( n ) } n = 1 ∞ have been studied by E. Cesaratto, B. Vallee, 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 E φ ( n ) , α . We give certain upper and lower bounds on the increasing speed of φ ( n ) when E φ ( n ) , α is of Hausdorff dimension 1 and a new class of sequences { φ ( n ) } n = 1 ∞ such that E φ ( n ) , α is of full dimension. For an irregular sequence { φ ( n ) } n = 1 ∞ , a full dimensional set E φ ( n ) , α is impossible.