MathDB

Let the base

2

representation of

x\in[0;1)

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}

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\, ?

Problem Statement