Metric properties of the product of consecutive partial quotients in continued fractions
- PDF / 482,437 Bytes
- 43 Pages / 429.408 x 636.768 pts Page_size
- 100 Downloads / 155 Views
METRIC PROPERTIES OF THE PRODUCT OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS∗ BY
Lingling Huang College of Information Science and Technology, Hunan Agricultural University, Changsha, 410128, P. R. China e-mail: lingling [email protected] AND
Jun Wu and Jian Xu∗∗ School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, P. R. China e-mail: [email protected], [email protected]
ABSTRACT
In the one-dimensional Diophantine approximation, by using the continued fractions, Khintchine’s theorem and Jarn´ık’s theorem are concerned with the growth of the large partial quotients, while the improvability of Dirichlet’s theorem is concerned with the growth of the product of consecutive partial quotients. This paper aims to establish a complete characterization on the metric properties of the product of the partial quotients, including the Lebesgue measure-theoretic result and the Hausdorff dimensional result. More precisely, for any x ∈ [0, 1), let x = [a1 , a2 , . . .] be its continued fraction expansion. The size of the following set, in the sense of Lebesgue measure and Hausdorff dimension, Em (ϕ):= {x ∈ [0, 1) : an (x) · · · an+m−1 (x) ≥ ϕ(n) for infinitely many n ∈ N}, are given completely, where m ≥ 1 is an integer and ϕ : N → R+ is a positive function. ∗ This work was supported by NSFC 11831007, 11571127. ∗∗ Corresponding author.
Received December 21, 2018 and in revised form august 23, 2019
1
2
L. HUANG, J. WU AND J. XU
Isr. J. Math.
1. Introduction It is well known that the continued fraction expansion of real numbers plays a significant role in studying the one-dimensional homogeneous Diophantine approximation. This can be inferred from the following two fundamental results. More specifically, let x = [a1 , a2 , . . .] be its continued fraction expansion and pn (x)/qn (x) be its nth convergent. Theorem 1.1 (Khintchine, [17]): • Optimal rational approximation of the convergent: pn (x) p min qx = qn−1 (x)x, (1.1) min , x − = x − 1≤q≤qn (x),p∈Z q qn (x) 1≤q 1, min qx ≤ 1/Q.
1≤q m. Together with a result of Khintchine [17] that there exists C > 1 such that for almost all x ∈ [0, 1), qn (x) ≤ C n
for all n 1,
we have Corollary 1.6: Let ϕ : N → [2, +∞) be a positive non-decreasing function and define Fm (ϕ) := {x ∈ [0, 1) : an (x) · · · an+m−1 (x) ≥ ϕ(qn (x)) for i.m. n ∈ N}. Then
⎧ ⎨0, if ∞ n=1 L(Fm (ϕ)) = ⎩1, if ∞
logm−1 ϕ(n) nϕ(n) logm−1 ϕ(n) n=1 nϕ(n)
< ∞, = ∞.
Corollary 1.6 recovers Khintchine’s theorem on the Lebesgue measure of W(φ) and Kleinbock and Wadleigh’s theorem on the measure of D(φ). More precisely, recall the relationship between continued fractions and the Diophantine approximation given above. When m = 1, the set Fm (ϕ) corresponds to the φ-well approximable set W(φ); when m = 2, it corresponds to the φ-Dirichlet nonimprovable set by a suitable choice of φ.
6
L. HUANG, J. WU AND J. XU
Isr. J. Math.
The Hausdorff dimension of Em (ϕ) is completely given in the following result: Theorem 1.7 (Hausdorff dimension
Data Loading...
