On the growth behavior of partial quotients in continued fractions

被引:3
|
作者
Shang, Lei [1 ]
Wu, Min [2 ]
机构
[1] Sun Yat sen Univ, Sch Math, Guangzhou 510275, Peoples R China
[2] South China Univ Technol, Sch Math, Guangzhou 510640, Peoples R China
基金
中国国家自然科学基金;
关键词
Continued fractions; Partial quotients; Residual sets; Hausdorff dimension;
D O I
10.1007/s00013-022-01821-2
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Let [a(1)(x), a(2)(x), a(3)(x), ...] be the continued fraction expansion of an irrational number x is an element of (0, 1). It is known that for Lebesgue almost all x is an element of (0, 1) \ Q, (lim inf )(n ->infinity) log a(n)(x)/ log n = 0 and (lim inf )(n ->infinity)log a(n)(x)/ log n = 1. In this note, the Baire classification and Hausdorff dimension of E(alpha, beta) : {x is an element of (0, 1) \ Q:(lim inf )(n ->infinity )log a(n)(x)/ log n=alpha, (lim inf )(n ->infinity) log a(n)(x)/ log n= beta} for all alpha, beta is an element of [0, infinity] with alpha <= beta are studied. We prove that E(alpha, beta) is residual if and only if alpha = 0 and beta = infinity, and the Hausdorff dimension of E(alpha, beta) is as follows: dim(H) E(alpha, beta) = {1, alpha=0; 1/2, alpha > 0.Moreover, the Hausdorff dimension of the intersection of E(alpha, beta) and the set of points with non-decreasing partial quotients is also provided.
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页码:297 / 305
页数:9
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