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

Abstract

This paper summarized recent achievements obtained by the authors about the box dimensions of the Besicovitch functions given by $$B(t): = \sum\limits_{k = 1}^\infty {\lambda _k^{s - 2} } \sin (\lambda _k t),$$ where 1<s<2, λ k > 0 tends to infinity as k→∞ and λ k satisfies λ k+1/λ k ≥ λ > 1. The results show that $$\mathop {\lim }\limits_{k \to \infty } \frac{{\log \lambda _{k + 1} }}{{\log \lambda _k }} = 1$$ is a necessary and sufficient condition for Graph(B(t)) to have same upper and lower box dimensions. For the fractional Riemann-Liouville differential operator Du and the fractional integral operator D−v, the results show that if λ is sufficiently large, then a necessary and sufficient condition for box dimension of Graph(D−v(B)), 0<v<s−1, to be s−v and box dimension of Graph(Du(B)), 0<u<2−s, to be s+u is also\(\mathop {\lim }\limits_{k \to \infty } \frac{{\log \lambda _{k + 1} }}{{\log \lambda _k }} = 1\).