The efficiency of approximating real numbers by Luroth expansion

For any x epsilon (0, 1], let x=1/d(1) + 1/d(1)(d(1)-1)d(2) + ... + 1/d(1)(d(1)-1)...d(n)-1(d(n-1) -1)d(n) +... be its Luroth expansion. Denote by P-n(x)/Q(n)(x) the partial sum of the first n terms in the above series and call it the nth convergent of x in the Luroth expansion. This paper is concerned with the efficiency of approximating real numbers by their convergents {P-n(x)/Q(n)(x)}(n >= 1) in the Luroth expansion. It is shown that almost no points can have convergents as the optimal approximation for infinitely many times in the Luroth expansion. Consequently, Hausdorff dimension is introduced to quantify the set of real numbers which can be well approximated by their convergents in the Luroth expansion, namely the following Jarnik-like set: {x epsilon (0, 1]: vertical bar x - P-n(x)/Q(n)(x)vertical bar < 1/Q(n)(x)(nu+1) infinitely often} for any nu >= 1.