For each real number $x \in (0,1)$, let $[a_1(x),a_2(x),\cdots , a_n(x),\cdots ]$ denote its continued fraction expansion. We study the convergence exponent defined by $\tau(x):= \inf\Big\{s \geq 0: \sum\limits_{n=1}^{\infty}\big(a_n(x)a_{n+1}(x)\big)^{-s}<\infty\Big\},$ which reflects the growth rate of the product of two consecutive partial quotients. As a main result, the Hausdorff dimensions of the level sets of $\tau(x)$ are determined.
Lulu FANG
,
Jihua MA
,
Kunkun SONG
,
Xin YANG
. MULTIFRACTAL ANALYSIS OF CONVERGENCE EXPONENTS FOR PRODUCTS OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS[J]. Acta mathematica scientia, Series B, 2024
, 44(4)
: 1594
-1608
.
DOI: 10.1007/s10473-024-0422-6
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