ON THE DISTRIBUTION OF THE RUDIN-SHAPIRO FUNCTION FOR FINITE FIELDS

Let q = p(r) be the power of a prime p and (beta(1), ..., beta(r)) be an ordered basis of F-q over F-p. For xi = Sigma(r)(j=1)x(j)beta(j) is an element of F-q with digits x(j) is an element of F-p, we define the Rudin-Shapiro function R on F-q by R(xi) = Sigma(r-1)(i=1)x(i)x(i+1), xi is an element of F-q. For a non-constant polynomial f (X) is an element of F-q[X] and c is an element of F-p we study the number of solutions xi is an element of F-q of R(f(xi)) = c. If the degree d of f(X) is fixed, r >= 6 and p -> infinity, the number of solutions is asymptotically p(r-1) for any c. The proof is based on the Hooley-Katz Theorem.