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 ≥ N 0 (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.