Abstract
AbstractLet 1 < s < 2, λk > 0 with λk → ∞ satisfy λk+1/λk ≥ λ > 1. For a class of Besicovich functions B(t) = $ \sum ^{\infty} _{k=1} \, \lambda ^{s-2} _{k} $ sin λkt, the present paper investigates the intrinsic relationship between box dimension of their graphs and the asymptotic behavior of {λk}. 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, dimBΓ(B) = dimBΓ(B) = s holds if and only if limn→∞ $ { { {\rm log} \lambda _{n+1}} \over { {\rm log} \lambda _{n} } } $ = 1. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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