The Growth Rate of Number Sequence in Engel Continued Fraction
For any real number x ∈[0,1),let x=[b1(x),b2(x),…]be its Engel continued fraction.To explore the growth rate of number sequence in Engel continued fraction expansion,this study mainly describes the Hausdorff dimension of the relevant exception set E(α,β)={x ∈[0,1):liminf n→∞ lnbn(x)/n=α,limsup n→∞ lnbn(x)/n=β｝when lnbn(x)in Engel continued fractional expansion grows at a linear rate.By constructing the Cantor subset of the exception set and using the mass distribution principle,the result that the exception set E(α,β)is full dimension for any0≤α≤β≤∞is given.This result complements the research on the growth rate of number sequences in Engel continued fraction expansion.
Engel continued fractionsHausdorff dimensionexceptional set
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