Abstract
Let \(\varvec{F}_q\) be the finite field of q elements, where \(q=p^r\) is a power of the prime p, and \(\left( \beta _1, \beta _2, \dots , \beta _r \right) \) be an ordered basis of \(\varvec{F}_q\) over \(\varvec{F}_p\). For
we define the Thue–Morse or sum-of-digits function \(T(\xi )\) on \(\varvec{F}_q\) by
For a given pattern length s with \(1\le s\le q\), a vector \(\varvec{\alpha }=(\alpha _1,\ldots ,\alpha _s)\in \varvec{F}_q^s\) with different coordinates \(\alpha _{j_1}\not = \alpha _{j_2}\), \(1\le j_1<j_2\le s\), a polynomial \(f(X)\in \varvec{F}_q[X]\) of degree d and a vector \(\mathbf{c} =(c_1,\ldots ,c_s)\in \varvec{F}_p^s\) we put
In this paper we will see that under some natural conditions, the size of \(\mathcal{T}(\mathbf{c} ,\varvec{\alpha },f)\) is asymptotically the same for all \(\mathbf{c} \) and \(\varvec{\alpha }\) in both cases, \(p\rightarrow \infty \) and \(r\rightarrow \infty \), respectively. More precisely, we have
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 \(1\le d<p\) we have the dichotomy that the bound is valid if \(s\le d\) and for \(s\ge d+1\) there are vectors \(\mathbf{c} \) and \(\varvec{\alpha }\) with \(\mathcal{T}(\mathbf{c} ,\varvec{\alpha },f)=\emptyset \) so that the bound fails for sufficiently large r. The case \(s=1\) was studied before by Dartyge and Sárközy.
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1 Introduction
1.1 The problem for binary sequences
For positive integers M and s, a binary sequence \((a_n)\) and a binary pattern
of length s we denote by \(N(a_n,M,\mathcal{E}_s)\) the number of n with \(0\le n<M\) and \((a_n,a_{n+1},\ldots ,a_{n+s-1})=\mathcal{E}_s.\) The sequence \((a_n)\) is normal if for any fixed s and any pattern \(\mathcal{E}_s\) of length s,
The Thue–Morse or sum-of-digits sequence \((t_n)\) is defined by
if
is the binary expansion of n. Recently, Drmota et al. [1] showed that the Thue–Morse sequence along squares, that is, \((t_{n^2})\) is normal. It is conjectured but not proved yet that the subsequence of the Thue–Morse sequence along any polynomial of degree \(d\ge 3\) is normal as well, see [1, Conjecture 1]. Even the weaker problem of determining the frequency of 0 and 1 in the subsequence of the Thue–Morse sequence along any polynomial of degree \(d\ge 3\) seems to be out of reach, see [1, above Conjecture 1].
However, the analog of the latter weaker problem for the Thue–Morse sequence in the finite field setting was settled by Dartyge and Sárközy [2].
1.2 The analog for finite fields
This paper deals with the following analog of the normality problem. Let \(q=p^r\) be the power of a prime p and
be an ordered basis of the finite field \(\varvec{F}_q\) over \(\varvec{F}_p\). Then any \(\xi \in \varvec{F}_q\) has a unique representation
The coefficients \(x_1,\ldots , x_r\) are called the digits with respect to the basis \(\mathcal{B}\).
Dartyge and Sárközy [2] introduced the Thue–Morse or sum-of-digits function \(T(\xi )\) for the finite field \(\varvec{F}_q\) with respect to the basis \(\mathcal{B}\):
Note that T is a linear map from \(\varvec{F}_q\) to \(\varvec{F}_p\). Actually, we can take any non-trivial linear map
from \(\varvec{F}_q\) to \(\varvec{F}_p\) without changing our results or proofs below, where the trace Tr is defined by (7).
For a given pattern length s with \(1\le s\le q\), a vector
with different coordinates, a polynomial \(f(X)\in \varvec{F}_q[X]\) and a vector \(\mathbf{c} =(c_1,\ldots ,c_s)\in \varvec{F}_p^s\) we put
In [2] the Weil bound, see Lemma 1, was used to bound the cardinality of \(\mathcal{T}(\mathbf{c} ,\varvec{\alpha },f)\) for \(s=1\):
Let \(f(X)\in \varvec{F}_q[X]\) be a polynomial of degree d. Then for all \(c \in \varvec{F}_p\)
where
Note that the condition \(\gcd (d,p)=1\) can be relaxed to the condition that f(X) is not of the form \(g(X)^p-g(X)+c\) for some \(g(X)\in \varvec{F}_q[X]\) and \(c\in \varvec{F}_q\). For example, \(f(X)=X^p\) is not of the form \(g(X)^p-g(X)+c\) but does not satisfy \(\gcd (d,p)=1\).
Our goal is to prove that, under some natural conditions, the size of \(\mathcal{T}(\mathbf{c} ,\varvec{\alpha },f)\) is asymptotically the same for all \(\mathbf{c} \) and \(\varvec{\alpha }\).
1.3 Results of this paper
First we study monomials and prove the following result in Sect. 4.
Theorem 1
Let d be any integer with \(1\le d<q\) with unique representation
where
Let denote by
the monomial of degree d.
1. For \(n\ge 2,\) assume
for some m and k with
For any positive integer
any vector \(\varvec{\alpha }\in \varvec{F}_q^s\) with \(\alpha _{j_1}\not = \alpha _{j_2}\) for \(1\le j_1<j_2\le s,\) and any \(\mathbf{c} \in \varvec{F}_p^s\) we have
2. Conversely, if
for any integer s with
there is a vector \(\varvec{\alpha }\in \varvec{F}_q^s\) with \(\alpha _{j_1}\not = \alpha _{j_2}\), \(1\le j_1<j_2\le s,\) and a vector \(\mathbf{c} \in \varvec{F}_p^s\) for which \(\mathcal{T}(\mathbf{c} ,\varvec{\alpha },f_d)\) is empty.
3. For any s with
and any vector \(\varvec{\alpha }\in \varvec{F}_q^s\) there is a vector \(\mathbf{c} \in \varvec{F}_p^s\) for which \(\mathcal{T}(\mathbf{c} ,\varvec{\alpha },f_d)\) is empty.
For \(d<p\) we have the following dichotomy:
Corollary 1
Assume \(1\le d<p\).
For \(s\le d\) we have for any vector \(\varvec{\alpha }\in \varvec{F}_q^s\) with \(\alpha _{j_1}\not = \alpha _{j_2}\), \(1\le j_1<j_2\le s,\) and any \(\mathbf{c} \in \varvec{F}_p^s\)
For s with \(q\ge s>d\) there is a vector \(\varvec{\alpha }\in \varvec{F}_q^s\) with \(\alpha _{j_1}\not = \alpha _{j_1}\), \(1\le j_1<j_2\le s,\) and a vector \(\mathbf{c} \in \varvec{F}_p^s\) for which \(\mathcal{T}(\mathbf{c} ,\varvec{\alpha },f_d)\) is empty.
Theorem 1 provides two asymptotic formulas for \(|\mathcal{T}(\mathbf{c} ,\varvec{\alpha },X^d)|\) for \(r\rightarrow \infty \) and \(p\rightarrow \infty \), respectively.
Assume that p, j, n, \(d=(d_0+d_1p+\cdots +d_{n-1} p^{n-1})p^j\) and s satisfying (2) are fixed. Then we have
for any vectors \(\mathbf{c} \in \varvec{F}_p^s\) and \(\varvec{\alpha }\in \varvec{F}_q^s\) with \(\alpha _{j_1}\not = \alpha _{j_2}\), \(1\le j_1<j_2\le s\). We may say that \(T(f_d)\) is r-normal if (2) is satisfied.
Assume that \(j=0\) and d, r and s are fixed with \(1\le s\le \min \{d,\lfloor (r-1)/2\rfloor \}\). Then we have
for any \(\mathbf{c} \in \varvec{F}_p^s\) and \(\varvec{\alpha }\in \varvec{F}_q^s\) with \(\alpha _{j_1}\not = \alpha _{j_2}\), \(1\le j_1<j_2\le s\). We may say that \(T(f_d)\) is p-normal for \(1\le s\le \min \{d,\lfloor (r-1)/2\rfloor \}\).
Theorem 1 is only non-trivial for small degrees. However, for very large degrees we prove the following non-trivial result in Sect. 5.
Theorem 2
Let \(f_{q-1-d}(X)=X^{q-1-d}\) be a monomial of degree \(q-1-d\) with \(1\le d< q-1\). Then for any \(\varvec{\alpha }\in \varvec{F}_q^s\) with \(\alpha _{j_1}\not = \alpha _{j_2}\), \(1\le j_1<j_2\le s,\) and any \(\mathbf{c} \in \varvec{F}_p^s,\) we have
Note that with the convention \(0^{-1}=0\) we have
and can identify the monomial \(f_{q-1-d}(X)=X^{q-1-d}\) with the rational function \(f_{-d}(X)=X^{-d}\). However, the latter representation is independent of q and we can state two asymptotic formulas for \(|\mathcal{T}(\mathbf{c} ,\varvec{\alpha },f_{-d})|\) as well.
For any fixed d, p and s we have
that is, \(T(f_{-d})\) is r-normal.
For any fixed d, s and r with \(1\le s\le \lfloor (r-1)/2\rfloor \) we have
that is, \(T(f_{-d})\) is p-normal for \(1\le s\le \lfloor (r-1)/2\rfloor \).
Finally, we extend our results to arbitrary polynomials in Sect. 6.
Theorem 3
Let d be any integer with \(1\le d<q\) and \(\gcd (d,q)=1\). Let \(f(X)\in \varvec{F}_q[X]\) be any polynomial of degree d.
1. Denote \(d_0\equiv d\bmod p,\) \(1\le d_0<p\). For any integer s with
any \(\varvec{\alpha }\in \varvec{F}_q^s\) with \(\alpha _{j_1}\not = \alpha _{j_2},\) \(1\le j_1<j_2\le s,\) and any \(\mathbf{c} \in \varvec{F}_p^s\) we have
2. Conversely, if \(f(X)\in \varvec{F}_p[X]\) and \(d<p,\) then for any integer s with
there is \(\varvec{\alpha }\in \varvec{F}_q^s\) with \(\alpha _{j_1}\not = \alpha _{j_2},\) \(1\le j_1<j_2\le s,\) and \(\mathbf{c} \in \varvec{F}_p^s\) for which \(\mathcal{T}(\mathbf{c} ,\varvec{\alpha },f)\) is empty.
3. For any \(f(X)\in \varvec{F}_q[X],\) any s with
and any \(\varvec{\alpha }\in \varvec{F}_q^s\) there is a vector \(\mathbf{c} \in \varvec{F}_p^s\) for which \(\mathcal{T}(\mathbf{c} ,\varvec{\alpha },f)\) is empty.
We give examples of degree d with \(\gcd (d,p)>1\) and \(\mathcal{T}(\mathbf{c} ,\varvec{\alpha },f)=\emptyset \) for any \(s\ge 1\) in Sect. 7.1.
Again, for \(f(X)\in \varvec{F}_p[X]\) and \(1\le d<p\) we have a dichotomy.
Moreover, for any fixed d, p and s with \(\gcd (d,q)=1\) and \(1\le s\le d_0\) and any \(f(X)\in \varvec{F}_p[X]\) of degree d, T(f) is r-normal. Note that any \(f(X)\in \varvec{F}_p[X]\) is an element of \(\varvec{F}_{p^{r}}[X]\) for \(r=1,2,\ldots \)
For fixed d, r and s with \(1\le s\le \min \{d,\lfloor (r-1)/2\rfloor \}\) and any \(f(X)\in \mathbf{Z} {[}X{]}\) of degree d, T(f) is p-normal. Here \(f(X)\in \mathbf{Z} {[}X{]}\) can be identified with an element of \(\varvec{F}_p[X]\) for all primes p.
We start with a section on preliminary results used in the proofs. Then we show that
under certain conditions in Sect. 3. In Sects. 4 to 6 we show that these conditions are fulfilled under the assumptions of our theorems. We finish the paper with some remarks on related work in Sect. 7.
2 Preliminary results
We start with the Weil bound, see [3, Theorem 5.38 and comments below], [4, Theorem 2E] or [5].
Lemma 1
Let \(\psi \) be the additive canonical character of the finite field \(\varvec{F}_q,\) and f(X) be a polynomial of degree \(d\ge 1\) over \(\varvec{F}_q,\) which is not of the form \(g(X)^p-g(X)+c\) for some polynomial \(g(X)\in \varvec{F}_q[X]\) and \(c\in \varvec{F}_q\). Then we have
We also use the analog of the Weil bound for rational functions
of Moreno and Moreno [6, Theorem 2]. We only need the special case that \(\deg (f)\le \deg (g)\).
Lemma 2
Let \(\psi \) be a nontrivial additive character of \(\varvec{F}_q\) and let \(\frac{f(X)}{g(X)}\in \varvec{F}_q(X)\) be a rational function over \(\varvec{F}_q\). Let s be the number of distinct roots of the polynomial g(X) in the algebraic closure \(\overline{\varvec{F}_q}\) of \(\varvec{F}_q\). Suppose that \(\frac{f(X)}{g(X)}\) is not of the form \(H(X)^p-H(X),\) where H(X) is a rational function over \(\overline{\varvec{F}_q}\). If \(\deg (f)\le \deg (g),\) then we have
Note that \(g(X)^p-g(X)+c\) with \(g(X)\in \varvec{F}_q(X)\) and \(c\in \varvec{F}_q\) can be written as \(h(X)^p-h(X)\) for \(h(X)=g(X)+\gamma \in \overline{\varvec{F}_q}(X),\) where \(\gamma \in \overline{\varvec{F}_q}\) is a zero of the polynomial \(X^p-X-c\).
Next we state Lucas’ congruence, see [7] or [8, Lemma 6.3.10].
Lemma 3
Let p be a prime. If m and n are two natural numbers with p-adic expansions
and
then we have
As a consequence of Lucas’ congruence we can count the number of nonzero binomials coefficients \({m\atopwithdelims ()n }\bmod p\) for fixed m. Indeed, by Lucas’ congruence
or equivalently,
Therefore, we have the following result of Fine [9, Theorem 2]:
Lemma 4
Let p be a prime and m an integer with p-adic expansion
Then the number of nonzero binomial coefficients \({m \atopwithdelims ()n}\bmod p\) with \(0\le n\le m\) is
3 Trace, dual basis and exponential sums
Let
denote the (absolute) trace of \(\xi \in \varvec{F}_q\). Let \((\delta _1,\ldots ,\delta _r)\) denote the (existent and unique) dual basis of the basis \(\mathcal{B} =( \beta _1,\ldots ,\beta _r)\) of \(\varvec{F}_q\), see for example [3], that is,
Then we have
and
Note that
since \(\delta _1,\ldots ,\delta _r\) are linearly independent. Note that we don’t have to restrict ourselves to this special choice of \(\delta \) and T but can deal with any non-trivial linear map
from \(\varvec{F}_q\) to \(\varvec{F}_p\).
Put
Since
we get
Separating the term for \(a_1=\cdots =a_s=0\) we get
where
denotes the additive canonical character of \(\varvec{F}_q\) and
If \(F_{a_1,\ldots ,a_s}(X)\) is not of the form \(g(X)^p-g(X)+c\) for any \((a_1,\ldots ,a_s)\ne (0,\ldots ,0)\), then the Weil bound, Lemma 1, can be applied and yields (6).
4 Monomials \(f_d(X)=X^d\)
Now we study the special case
Put \(\varvec{\alpha }^k=(\alpha _1^k,\ldots ,\alpha _s^k)\). Since \((X+\alpha )^{dp^j}=(X^{p^j}+\alpha ^{p^j})^d\) and \(\xi \mapsto \xi ^{p^j}\) permutes \(\varvec{F}_q\) we have
and we may assume \(j=0\). Since
we may restrict ourselves to the case \(d<q\).
To prove the first part of Theorem 1 we have to show that (6) is applicable. By (9) with
we have
and thus
Assume that for some \((a_1,\ldots ,a_s)\in \varvec{F}_p^s\setminus \{(0,\ldots ,0)\}\) we have
for some polynomial \(g(X)\in \varvec{F}_q[X]\) and some constant \(c\in \varvec{F}_q\). We have
and
Then either
Let
be the p-adic expansion of d. Assume that there are \(k\ge 0\) consecutive digits
of maximal size and
Note that \(\deg (F_{a_1,\ldots ,a_s})\le d-d_0\) by (11) with the convention \(\deg (0)=-1\). In both cases, (12) and (13), the coefficients of \(F'_{a_1,\ldots ,a_s}(X)\) at \(X^{d-1-\ell }\) are zero for \(\ell =0,\ldots ,d-(d-d_0)/p-1\). Since \(\delta d\not = 0\) we get from (10)
By Lucas’ congruence, Lemma 3, we have
as well as
since
for some
and
for some
and any \(0\le \ell \le (d_{m+k}+1)p^k-1\).
Note that
Combining (14) with (15) and (16), respectively, we get
and
respectively.
Hence, if \(s\le d_0\) (\(n=1\) or \(k=0\)) or \(s\le (d_{m+k}+1)(p^k-p^{k-1})\) (\(n\ge 2\) and \(k\ge 1\)), the \(s \times s\) coefficient matrix of the equations for \(\ell =0,\ldots ,s-1\) of (17) or (18), respectively, is an invertible Vandermonde matrix and we get
contradicting \((a_1,\ldots ,a_s)\in \varvec{F}_p^s\setminus \{(0,\ldots ,0)\}\). For the second case we used that \(\xi \mapsto \xi ^{p^m}\) permutes \(\varvec{F}_q\) and the \(\alpha _i^{p^m}\), \(i=1,\ldots ,s\), are pairwise distinct.
Proof of the second part of Theorem 1: now assume \(d<p^n\) for some n with \(1\le n\le r\), that is, \(d_n=\cdots =d_{r-1}=0\), and assume (3) and (4). Let D be the number of binomial coefficients \({d\atopwithdelims ()\ell }\), \(\ell =1,\ldots ,d\), which are nonzero modulo p. By Lemma 4 we have
For any \(\alpha \in \varvec{F}_q\) the polynomial
is in the vector space generated by the monomials \(X^{d-\ell }\) with nonzero \({d\atopwithdelims ()\ell }\bmod p\), \(\ell =0,\ldots ,d-1\), of dimension D. For \(D<s\le q\) and any \((\alpha _1,\ldots ,\alpha _s)\in \varvec{F}_q^s\) there is a nontrivial linear combination
of the zero polynomial with \((\rho _1,\ldots ,\rho _s)\in \varvec{F}_q^s\setminus \{(0,\ldots ,0)\}\). If \(D<s\le p\) and we take \(\alpha _i\in \varvec{F}_p\), \(i=1,\ldots ,s\), then we may assume \(\rho _i=a_i\in \varvec{F}_p\) and
Taking \((a_1,\ldots ,a_s)\in \varvec{F}_p^s\setminus \{(0,\ldots ,0)\}\) from the previous step, the vector space of solutions \((c_1,\ldots ,c_s)\in \varvec{F}_p^s\) of the equation
is of dimension \(s-1\). More precisely, the mapping
is surjective since \((a_1,\ldots ,a_s)\) is not the zero vector. By the rank-nullity theorem its kernel is of dimension \(s-1\).
That is, not all \(\mathbf{c} =(c_1,\ldots ,c_s)\in \varvec{F}_p^s\) are attained as
for any \(\xi \in \varvec{F}_q\), namely those \(\mathbf{c} \) which are not in the kernel of \(\varphi \). We can extend this argument to \(s>p\) by extending \((a_1,\ldots ,a_p)\in \varvec{F}_p^p\setminus \{(0,\ldots ,0)\}\) to \((a_1,\ldots ,a_p,0,\ldots ,0)\in \varvec{F}_p^s\setminus \{(0,\ldots ,0)\}\).
Proof of the third part of Theorem 1: now we drop the condition (3) but then s has to satisfy the stronger condition (5) instead of (4). We extend the definition of the trace to polynomials \(f(X)\in \varvec{F}_{p^r}[X]\),
For each \(\alpha \in \varvec{F}_q\) we have
since the trace is \(\varvec{F}_p\)-linear, and thus it lies in the \(\varvec{F}_p\)-linear space generated by the polynomials \(\mathrm{Tr}(\beta _i X^{d-\ell })\) with nonzero \({d\atopwithdelims ()\ell }\) modulo p, \(i=1,\ldots ,r\), \(\ell =1,\ldots ,d\), of dimension at most Dr, where \(\{\beta _1,\ldots ,\beta _r\}\) is a basis of \(\varvec{F}_q\) over \(\varvec{F}_p\). Now let \(s> Dr\), then for any \(\varvec{\alpha }=\left( \alpha _1,\dots \alpha _s \right) \in \varvec{F}_q^s\) consider the set of polynomials
Since \(s>Dr\) there is a nontrivial \(\varvec{F}_p\)-linear combination
of the zero polynomial. Now consider the linear subspace of solutions \((c_1,\ldots ,c_s)\in \varvec{F}_p^s\) of the equation \(a_1c_1+\cdots +a_sc_s=0\) which is of dimension \(s-1\). Let \(\mathbf{c} \in \varvec{F}_p^s\) be a point which does not lie in this linear subspace, then \(\mathbf{c} \) is not attained as \(\mathbf{c} =(\mathrm{Tr}(\delta ((\xi +\alpha _i)^d-\alpha _i^d)))_{i=1}^s\) for any \(\xi \in \varvec{F}_q\).
5 Rational functions \(f_{-d}(X)=X^{-d}\)
Let \(f_{q-d-1}(X)=X^{q-1-d}\) be a monomial of degree \(q-d-1\), where \(1\le d< q-1\). With the convention \(0^{-1}=0\) we can identify \(f_{q-d-1}(X)\) with the rational function \(f_{-d}(X)=X^{-d}\). Let \(\gcd (d,q)=p^j\). Since
and \(\xi \mapsto \xi ^{p^j}\) permutes \(\varvec{F}_q\) we have
and may restrict ourselves to the case \(\gcd (d,q)=1\), that is,
We first show that there is no nonzero s-tuple
such that
for any rational function \(H(X)\in \overline{\varvec{F}_p}(X)\). We have
where
and
Suppose to the contrary that there exists a rational function
satisfying
Therefore, we have
Clearing denominators we obtain
and thus \(v(X)^p\) divides g(X), hence
Now by taking derivatives of both sides of (19) and clearing denominators we get
Without loss of generality we may assume \(a_1\ne 0\), thus
and
Moreover, \((X+\alpha _1)^{d-1}\) and \((X+\alpha _1)^d\) are the largest powers dividing \(g'(X)\) and g(X), respectively, that is,
is the largest power of \((X+\alpha _1)\) dividing the left hand side of (20). Observing that \(g(X)^2\) and thus the right hand side of (20) is divisible by
we get
and thus
which is a contradiction.
We showed that the conditions of Lemma 2 are satisfied and Theorem 2 follows from (8) and Lemma 2 since
where \(-\varvec{\alpha }=\{-\alpha _1,\ldots ,-\alpha _s\}\).
6 Arbitrary polynomials
In this section we prove Theorem 3.
Let
be a polynomial of degree
Proof of the first part: we have to show that (6) is applicable, that is, the polynomial \(F_{a_1,\ldots ,a_s}(X)\) defined by (9) is not of the form \(g(X)^p-g(X)+c\) for any \((a_1,\ldots ,a_s)\ne (0,\ldots ,0)\).
Suppose the contrary that there exists an s-tuple
such that the polynomial
can be written as
We have either
or
Hence,
where we used the convention \(\deg (0)=-1\). We conclude that the coefficients \(\delta R_\ell \) of \(F_{a_1,\dots ,a_s}(X)\) at \(X^{\ell }\) vanish for \(\ell =d-d_0+1,\dots , d\). Since \(\delta \not = 0\) we have
Note that by Lucas’ congruence, Lemma 3,
Define \(T_\ell \), \(\ell =0,\ldots d_0-1\), recursively by
and
for \(\ell =1, \ldots , d_0-1\). Next we show that
For \(\ell =0\) the formula follows from (21) and for \(\ell =1,\ldots ,d_0-1\) from (23) we get by induction
and from (21)
Moreover, we get
from (21), (23) again by induction.
By (24) and (22) we get since \(\gamma _d\ne 0\),
Thus for \(s\le d_0\), the \((s\times s)\)-coefficient matrix
of the system of the first s equations is a regular Vandermonde matrix and we get \((a_1,\ldots ,a_s)=(0,\ldots ,0)\), which is a contradiction.
For the second part of Theorem 3 we assume \(f(X)\in \varvec{F}_p[X]\) and notice that for any \(\alpha \in \varvec{F}_q\) the element \(f(X+\alpha )-f(\alpha )\) is in the vector space generated by the monomials \(X^i\), \(i=1,\ldots ,d\), of dimension d.
If \(d<s\le p\), we can choose any \(\varvec{\alpha }\in \varvec{F}_p^s\). Then
are linearly dependent over \(\varvec{F}_p\) as well as
that is,
for some \((a_1,\ldots ,a_s)\in \varvec{F}_p^s\setminus \{(0,\ldots ,0)\}\) and the result follows since not all \((c_1,\ldots ,c_s)\in \varvec{F}_p^s\) satisfy \(a_1c_1+\cdots +a_sc_s=0\).
If \(d<p\) and \(s>p\), we can choose \((a_1,\dots , a_p) \in \varvec{F}_p^p \setminus \{(0,\ldots ,0)\}\) as in the case \(s=p\) and extend it to \((a_1,\dots , a_p,a_{p+1},\ldots ,a_s)\in \varvec{F}^s_p \setminus \{(0,\ldots ,0)\}\) with \(a_{p+1}=\cdots =a_s=0\).
Proof of the third part of Theorem 3: recall that \(\{\beta _1,\ldots ,\beta _r\}\) is a basis of \(\varvec{F}_q\) over \(\varvec{F}_p\). Each \(\delta (f(X+\alpha )-f(\alpha ))\) lies in the \(\varvec{F}_p\)-vector space generated by
of dimension dr. The dimension of the vector space generated by
is at most dr. If \(q\ge s>dr\), there is a nontrivial linear combination
for any \(\varvec{\alpha }\in \varvec{F}_q^s\) and the result follows.
7 Final remarks
7.1 Examples for \(\gcd (d,p)>1\) and \(\mathcal{T}(\mathbf{c} ,\alpha ,f)=\emptyset \)
Now we provide an example that if we drop the condition on s in part 1 of Theorem 3, the restriction \(\gcd (d,q)=1\), that is \(d_0\ge 1\), is needed.
Choose any f(X) of the form
Then we obtain
for all \(\xi \in \varvec{F}_q\), that is, any vector \((c_1,\ldots ,c_r)\in \varvec{F}_p^r\) with \(c_1\ne \mathrm{Tr}(c)\) is not attained as \((T(f(\xi +\alpha _i)))_{i=1}^s\).
We conclude that for polynomials of degree d with \(\gcd (d,p)>1\), the bound of Theorem 3 may not hold for all s. However, by Theorem 1, for monomials the restriction \(\gcd (d,p)=1\) is not needed.
7.2 Missing digits and subsets
For subsets \(\mathcal{D}\) of \(\varvec{F}_p\), the closely related problem of estimating the number of \(\xi \in \varvec{F}_q\) with
was studied in [10,11,12], that is, \(f(\xi )\) ’misses’ the digits in \(\varvec{F}_p\setminus \mathcal{D}\). It is straightforward to extend these results combining our approach with certain bounds on character sums to estimate the number of \(\xi \in \varvec{F}_q\) with
For example, for \(\mathcal{D}=\{0,\ldots ,t-1\}\) we can use the bound on exponential sums of [13].
Instead of restricting the set of digits we may restrict the set of \(\xi \). That is, for a subset \(\mathcal{S}\) of \(\varvec{F}_q\) we are interested in the number of solutions \(\xi \in \mathcal{S}\) of
for any fixed \(\mathbf{c} \in \varvec{F}_p^s\). Typical choices of \(\mathcal{S}\) are ‘boxes’ [13, 14] and ’consecutive’ elements [15].
7.3 Optimality and prescribed digits
Swaenepoel [16] improved the bound (1) of [2] in the case when the polynomial f(X) has degree 2 or is a monomial. In particular, for \(s=1\) and \(d=2\) the improved bound of [16] is optimal. She also generalized (1) to several polynomials with \(\varvec{F}_p\)-linearly independent leading coefficients [16, Theorem 1.5].
Moreover, in [17] Swaenepoel studied the number of solutions \(\xi \in \varvec{F}_q\) for which some of the digits of \(f(\xi )\) are prescribed, that is, for given \(\mathcal{I}\subset \{1,\ldots ,r\}\) and given \(c_i\in \varvec{F}_p\), \(i\in \mathcal{I}\), the number of \(\xi \in \varvec{F}_q\) with
7.4 Related work on pseudorandom number generators
Some of the ideas of the proofs in this paper are based on earlier work on nonlinear, in particular, inversive pseudorandom number generators, see [18,19,20].
More precisely, in [20] the q-periodic sequence \((\eta _n)\) over \(\varvec{F}_q\) defined by
passes the s-dimensional lattice test if s polynomials of the form
are \(\varvec{F}_q\)-linearly independent. However, in the proofs of this paper we need that they are linearly independent (resp. dependent) over \(\varvec{F}_p\).
To prove Theorem 2 for \(d<p\), the method of [19] can be easily adjusted using [19, Lemma 2]. However, for \(d\ge p\) we had to use a different approach since [19, Lemma 2] is not applicable in this case.
Finally, in the proof of [18, Theorem 4] we showed that polynomials of the form \(F_{a_1,\ldots ,a_s}(X)\) can only be identical 0 if \(a_1=\cdots =a_s=0\). However, in the proof of Theorem 3 we had to show that \(F_{a_1,\ldots ,a_s}(X)\) is not of the form \(g(X)^p-g(X)+c\) and we had to modify the idea of [18].
7.5 Rudin–Shapiro function
The Rudin–Shapiro sequence \((r_n)\) is defined by
if
Müllner showed that the Rudin–Shapiro sequence along squares \((r_{n^2})\) is normal [21].
The Rudin–Shapiro function \(R(\xi )\) for the finite field \(\varvec{F}_q\) with respect to the ordered basis \((\beta _1,\ldots ,\beta _r)\) is defined as
For \(f(X)\in \varvec{F}_q[X]\) and \(c\in \varvec{F}_p\) let
It seems to be not possible to use character sums to estimate the size of \(\mathcal{R}(c,f)\). However, in [22] the Hooley–Katz Theorem, see [23, Theorem 7.1.14] or [24] was used to show that if \(d=\deg (f)\ge 1\),
where \(C_{r,d}\) is a constant depending only on r and d. In particular, we have for fixed d and \(r\ge 6\),
that is, R(f) is p-normal for \(s=1\) and \(r\ge 6\).
However, we are not aware of a result on the r-normality of R(f).
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The authors are partially supported by the Austrian Science Fund FWF Project P 30405. They wish to thank the anonymous referee for very useful suggestions.
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Makhul, M., Winterhof, A. Normality of the Thue–Morse function for finite fields along polynomial values. Res. number theory 8, 38 (2022). https://doi.org/10.1007/s40993-022-00335-8
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DOI: https://doi.org/10.1007/s40993-022-00335-8