SCHWEITZER 2017 Problem 8

Let the base $2$ representation of $x\in[0;1)$ be $x=\sum_{i=0}^\infty \frac{x_i}{2^{i+1}}$ . (If $x$ is dyadically rational, i.e. $x\in\left\{\frac{k}{2^n}\,: k,n\in\mathbb{Z}\right\}$ , then we choose the finite representation.) Define function $f_n:[0;1)\to\mathbb{Z}$ by $$f_n(x)=\sum_{j=0}^{n-1}(-1)^{\sum_{i=0}^j x_i}.$$ Does there exist a function $\varphi:[0;\infty)\to[0;\infty)$ such that $\lim_{x\to\infty} \varphi(x)=\infty$ and $$\sup_{n\in\mathbb{N}}\int_0^1 \varphi(|f_n(x)|)\mathrm{d}x<\infty ?$$