Let Fq be a finite field with q elements, Fq((z−1)) denote the field of all formal Laurent series with coefficients in Fq and I be the valuation ideal of Fq((z−1)). For any formal Laurent series , the series 852cb85b5a19b9e8d3d73d05e"> is the Oppenheim expansion of x . Suppose ϕ:N→R+ is a function satisfying ϕ(n)/n→∞ as n→∞. In this paper, we quantify the size, in the sense of Hausdorff dimension, of the set
where Δ0(x)=dega1(x) and Δn(x)=degan+1(x)−2degan(x)−degrn(an(x))+degsn(an(x)) for all n≥1. As applications, we investigate the cases when 8066f50fe905406d93db9af4" title="Click to view the MathML source">ϕ(n) are the given polynomial or exponential functions. At the end of the article, we list some special cases (including Lüroth, Engel, Sylvester expansions of Laurent series and Cantor infinite products of Laurent series) to which we apply the conclusions above.