Hausdorff dimension of some exceptional sets in L\"{u}roth expansions

In this paper, we study the metrical theory of the growth rate of digits in L\"{u}roth expansions. More precisely, for x∈ ( 0,1 ], let [ d₁( x ) ,d₂( x ) ,⋯ ] denote the L\"{u}roth expansion of x, we completely determine the Hausdorff dimension of the following sets {align*} Eₛᵤₚ( ψ ) =\{ x∈ ( 0,1 ] :n→ ∞log d_n( x )/ψ ( n )=1 \} , {align*} {align*} E( ψ ) =\{ x∈ ( 0,1 ] :limn→ ∞log d_n( x )/ψ ( n )=1 \} {align*} and {align*} Einf( ψ ) =\{ x∈ ( 0,1 ] : n→ ∞log d_n( x )/ψ ( n )=1 \} , {align*} where ψ :N → R ⁺ is an arbitrary function satisfying ψ ( n ) → ∞ as n→ ∞.