Motivation. When we try to construct a (pseudo-)random sequence $s:\newcommand{\N}{\mathbb{N}}\N\to\{0,1\}$ we often want $s$ itself, and some of its subsequences, to be normal.
Question. Is there a computable binary sequence $s:\N\to\{0,1\}$ such that for every strictly increasing polynomial $p:\N\to\N$ the binary sequence $s\circ p:\N\to\{0,1\}$ is normal?
Any ptime-random computable sequence works.
A sequence is polytime random if no betting strategy computable in polynomial time can win unlimited amounts of money on it. If there exists some strictly increasing polynomial $p$ such that $s \circ p$ is not normal, we can built a polytime betting strategy from $p$ and a finite subsequence occuring "too often" in $s \circ p$; this shows that $s$ was not polytime random.
Constructing a computable polytime random sequence is just a matter of diagonalization.
