HAUSDORFF DIMENSION OF THE GRAPHS OF SOME PERTURBED RADEMACHER SERIES

Abstract

For , let ${\text{Ω}}_{\alpha }={\displaystyle \prod \left[2^{-\alpha }-a_i,2^{-\alpha }+a_i\right]}$ be the probability space of Kahane and Salem, where $\left\{a_i\right\}$ is a positive sequence satisfying certain decaying condition. Let ${\left\{R_i\right\}}_1^{\infty }$ be the sequence of Rademacher functions. We show that for each $\alpha \in (0,1)$ and for almost all $\left\{{\mathrm{\xi }}_i\right\}\in {\mathrm{\Omega }}_{\alpha }$, the graph of $f(x)={\displaystyle \sum _{i=1}^{\infty }{\mathrm{\xi }}_1\cdots {\mathrm{\xi }}_i{R}_i(x)}$ has Hausdorff dimension $2-\alpha$.