2024/08/31 by Fang, Lulu, Moreira, Carlos Gustavo, Zhang, Yiwei · 2 citations
#11K50 #28A80 #37D35 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2409.00521
In 1928, Jarn'ık \citeJar obtained that the set of continued fractions with bounded coefficients has Hausdorff dimension one. Good \citeGoo observed a dimension drop phenomenon by proving that the Hausdorff dimension of the set of continued fractions whose coefficients tend to infinity is one-half. For the set of continued fractions whose coefficients tend to infinity rapidly, Luczak \citeLuc and Feng et al. \citeFWLT showed that its Hausdorff dimension decreases even further. Recently, Liao and Rams \citeLR16 also observed an analogous dimension drop phenomenon when they studied the subexponential growth rate of the sum of coefficients. In this paper, we consolidate and considerably extend the studies of the abovementioned problem into a general dimension drop problem on the distribution of continued fractions with large coefficients. As applications, we use a different approach to reprove a result of Wang and Wu on the dimensions of the Borel-Bernstein sets \citeWW, fulfil the dimension gap proposed by Liao and Rams \citeLR16, and establish several new results concerning the dimension theory of liminf and limsup sets related to the maximum of coefficients.