On digital blocks of polynomial values and extractions in the Rudin–Shapiro sequence

Let P(x) ∈ ℤ [x] be an integer-valued polynomial taking only positive values and let d be a fixed positive integer. The aim of this short note is to show, by elementary means, that for any sufficiently large integer N ≥ N0(P,d) there exists n such that P(n) contains exactly N occurrences of the block (q − 1, q − 1,..., q − 1) of size d in its digital expansion in base q. The method of proof allows to give a lower estimate on the number of “0” resp. “1” symbols in polynomial extractions in the Rudin–Shapiro sequence