Metrical properties of products formed by consecutive partial quotients in Lüroth expansions

Yan Feng and Fuming Jiang

Chaos, Solitons & Fractals, 2025, vol. 191, issue C

Abstract: For x∈(0,1], let [d1(x),…,dk(x),…] be its Lüroth expansion, and {pk(x)qk(x)}k≥1 be its convergents. Brown-Sarre et al. considered the metrical properties of products derived from sequential partial quotients raised to different powers. More precisely, for any t=(t0,…,tm)∈R+m+1, they examined the metrical characteristics of the set Et(ψ)={x∈(0,1]:dkt0(x)dk+1t1(x)⋯dk+mtm(x)≥ψ(k)for i.mk},where m represents a positive integer, ψ:N→(1,∞) is a function. In this paper, we will proceed with the study of these issues. Namely, the primary aim of this research is to explore the metrical properties of the following sets Dt(ψ)={x∈(0,1]:dkt0(x)dk+1t1(x)⋯dk+mtm(x)≥ψ(k),k≥1}and Ft(ψ)={x∈(0,1]:dkt0(x)dk+1t1(x)⋯dk+mtm(x)≥ψ(qk(x))for i.m.k}.Moreover, we also give due consideration to the Hausdorff dimension of the exceptional set associated with the convergence exponent cm(x)=inf{l≥0:∑k=1∞(dk(x)dk+1(x)⋯dk+m(x))−l<∞}.

Keywords: Lüroth expansion; Hausdorff dimension; Product of partial quotients (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:191:y:2025:i:c:s0960077924013407

DOI: 10.1016/j.chaos.2024.115788

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