Normality of the Thue–Morse function for finite fields along polynomial values

• Published in: Research in number theory Vol. 8; no. 3
• Main Authors: Makhul, Mehdi, Winterhof, Arne
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2022; Springer Nature B.V
• Subjects: Fields (mathematics); Mathematics; Mathematics and Statistics; Number Theory; Polynomials; Finite fields; 11T06; Thue–Morse function; 11L99; 11A63; Sum of digits; Normality; Exponential sums; Polynomial equations; 11T23
• ISSN: 2522-0160; 2363-9555
• DOI: 10.1007/s40993-022-00335-8

Let F q be the finite field of q elements, where q = p r is a power of the prime p , and β 1 , β 2 , ⋯ , β r be an ordered basis of F q over F p . For ξ = ∑ i = 1 r x i β i , x i ∈ F p , we define the Thue–Morse or sum-of-digits function T ( ξ ) on F q by T ( ξ ) = ∑ i = 1 r x i . For a given pattern length s with 1 ≤ s ≤ q , a vector α = ( α 1 , … , α s ) ∈ F q s with different coordinates α j 1 ≠ α j 2 , 1 ≤ j 1 < j 2 ≤ s , a polynomial f ( X ) ∈ F q [ X ] of degree d and a vector c = ( c 1 , … , c s ) ∈ F p s we put T ( c , α , f ) = { ξ ∈ F q : T ( f ( ξ + α i ) ) = c i , i = 1 , … , s } . In this paper we will see that under some natural conditions, the size of  T ( c , α , f ) is asymptotically the same for all  c and α in both cases, p → ∞ and r → ∞ , respectively. More precisely, we have | T ( c , α , f ) | - p r - s ≤ ( d - 1 ) q 1 / 2 under certain conditions on d ,  q and s . For monomials of large degree we improve this bound as well as we find conditions on d ,  q and s for which this bound is not true. In particular, if 1 ≤ d < p we have the dichotomy that the bound is valid if s ≤ d and for s ≥ d + 1 there are vectors c and  α with T ( c , α , f ) = ∅ so that the bound fails for sufficiently large r . The case s = 1 was studied before by Dartyge and Sárközy.