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

✍ Scribed by S. P. Zhou; G. L. He

• Publisher: John Wiley and Sons
• Year: 2005
• Tongue: English
• Weight: 111 KB
• Volume: 278
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310267

✦ Synopsis

Abstract

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) = $ \sum ^{\infty} _{k=1} , \lambda ^{s-2} _{k} $ sin λ~k~t, 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~B~Γ(B) = dim~B~Γ(B) = s holds if and only if lim~n→∞~ $ { { {\rm log} \lambda _{n+1}} \over { {\rm log} \lambda _{n} } } $ = 1. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)