A note on Engel series expansions of Laurent series and Hausdorff dimensions

Let FqFq be the finite field with q   elements and Fq((z−1))Fq((z−1)) be the field of all formal Laurent series with coefficients in FqFq. For any x∈I:=z−1Fq((z−1))x∈I:=z−1Fq((z−1)), the Engel series expansion of x   is ∑n=1∞1a1(x)⋯an(x) with aj(x)∈Fq[z]aj(x)∈Fq[z]. Suppose that ϕ:N→R+ϕ:N→R+ is a function satisfying ϕ(n)⩾nϕ(n)⩾n for all integers n large enough. In this note, we consider the following setE(ϕ)={x∈I:limn→∞degan(x)ϕ(n)=1}, and establish a lower bound of its Hausdorff dimension. As a direct application, we obtain in particular dimH{x∈I:limn→∞degan(x)nβ=γ}=1 (where β>1β>1, γ>0γ>0 or β=1β=1, γ⩾1γ⩾1, and dimHdimH denotes the Hausdorff dimension), which generalizes a result of J. Wu dated 2003.