The convergence exponent and the well approximated sets for Lüroth expansion

Abstract

For any x in (0,1], let [d1(x),d2(x),⋯,dn(x),⋯] be its Lüroth expansion and {pn(x)qn(x)}n≥1 its convergent sequence. In this note, we study the multifractal spectrum of the convergence exponent of the sequence {dn(x)}n≥1 defined byτ(x):=inf⁡{s≥0:∑n≥1dn−s(x)<∞} and determine the Hausdorff dimension of the setE(α,β)={x∈(0,1]:τ(x)=α}⋂Jβ for any α≥0,β≥0, whereJβ={x∈(0,1]:|x−pn(x)qn(x)|<1qn(x)β+1for infinitely manyn≥1}.