On a class of Besicovitch functions to have exact box dimension: A necessary and sufficient condition

Let 1 < s < 2, λ_k > 0 with λ_k → ∞ satisfy λ_k+1/λ_k ≥ λ > 1. For a class of Besicovich functions B(t) = sin λ_kt, the present paper investigates the intrinsic relationship between box dimension of their graphs and the asymptotic behavior of {λ_k}. We show that the upper box dimension does not exceed s in general, and equals to s while the increasing rate is sufficiently large. An estimate of the lower box dimension is also established. Then a necessary and sufficient condition is given for this type of Besicovitch functions to have exact box dimensions: for sufficiently large λ, dim _BΓ(B) = dim _BΓ(B) = s holds if and only if lim_n→∞ = 1. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)