On the Hausdorff dimension of average type sums of Rademacher functions.

Abstract

We prove that for any $a, c \in (0,1)$ and any $b,d \in {\mathbb R},$ the Hausdorff dimension of $ \{ x\in [0,1] : n^{-a} \sum_{j=1}^{n}r_j(x) \to b \hbox{ and } n^{-c} \sum_{j=1}^{n} r_j(x) r_{j+1}(x)\to d\},$ is equal to 1, where $\{r_n(x)\}_{n \geq 1},$ are the Rademacher functions. We give also an extension of this result.