aa r X i v : . [ m a t h . N T ] F e b Partial Gaussian Sums in Finite Fields
Ke Gong
Department of Mathematics, Henan UniversityKaifeng 475004, P. R. China [email protected]
Abstract
We generalize Burgess’ results on partial Gaussian sums to arbitrary finite fields.The main ingredients are the classical method of amplification, two deep results onmultiplicative energy for subsets in finite fields which are obtained respectively by thetools from additive combinatorics and geometry of numbers, and a technique of Chamizofor treating the difficulty caused by additive character. Our results include the recentworks on character sums in finite fields by M.-C. Chang and S. V. Konyagin.
Let p be a prime, χ a non-principal character modulo p . We denote e p ( y ) := exp(2 πiy/p ) asusual. Sums of the form N + H X x = N χ ( x ) e p ( ax ) , (1)are often encountered in analytic number theory.We call the sums (1) pure character sums if a ≡ p ), otherwise mixed charactersums. If H = p we say the sums (1) complete , otherwise incomplete (or partial as Burgessused).In the case of a p ) and H < p , sums (1) are usually called partial Gaussiansums, which have been well studied by Vinogradov [18] and Burgess [5]. In this paper we tryto generalize Burgess’ results to arbitrary finite fields.By a well-known generalization of the P´olya-Vinogradov inequality we have N + H X x = N χ ( x ) e p ( ax ) ≪ p / log p. r we have N + H X x = N χ ( x ) ≪ H − /r p ( r +1) / r log p. (2)Fifteen years later, by a modification of his method in proving (2), Burgess [5] proved thefollowing estimates for general partial Gaussian sums. Theorem 1.
Let χ be a non-principal character modulo a prime p . Then for any integers r ≥ , a , N and ≤ H < p we have N + H X x = N χ ( x ) e p ( ax ) ≪ H − /r p / r − log p. (3)On the other hand, parallel to the pure character sums (2) in prime field F p , there arealso many works on pure character sums in general finite fields F q , q = p n . See the papers ofDavenport and Lewis [9], Chang [7] and Konyagin [13]. So it is naturally to consider partialGaussian sums in arbitrary finite fields. However, such a generalization is quite unusuallybecause the additive character e p ( · ) causes additional difficulty even in the case of prime field.Indeed Burgess himself has remarked that the argument used to obtain (2) depended on thesummand being multiplicative (see Burgess [5, p. 589]). Thus the method used by Burgessdoes not have any natural extensions to the case of arbitrary finite fields. And even nowadays,although the results we obtain in this paper match Burgess’ results in the same range, theyare not as explicit as those of Burgess.Recently, Chamizo [6] presented a new proof of Burgess’ partial Gaussian sums on theThird Conference on Number Theory at University of Salamanca (Salamanca, July 2009).Chamizo’s used essentially the classical method of amplification in the form of Iwaniec andKowalski [11]. He ingeniously introduced a trick to overcome the difficulty caused by additivecharacter.In the present paper we generalize Burgess’ partial Gaussian sums to arbitrary finitefields. Two deep results on multiplicative energy for subsets in finite fields, which are obtainedrespectively by some tools from additive combinatorics and geometry of numbers, are involvedhere. We will also use Chamizo’s trick.We finally remark that Perel’muter [15] has studied partial Gaussian sums over additivesubgroup of F p n . However he mainly concerned with the algebraic respects. The method of amplification was first used in number theory by Vinogradov [19], then introduced byKaratsuba [12] into the study of character sums. Now it is a classical method, see Friedlander [10], Iwaniecand Kowalski [11], Chang [7]. Notation
Throughout the paper we will use the following notations.Let p be an odd prime, q an integer with q = p n , and F p the prime field. Let F q denotethe finite field with q elements.We recall that the function Tr( z ) = n − X i =0 z p i is called the trace of z ∈ F p n over F p .Define e p ( z ) = exp(2 πiz/p ). Then the set of functions ψ a ( z ) = e p (Tr( az )), a ∈ F p n , formthe set of additive characters of F p n , with ψ being the trivial character.Let χ be a nontrivial multiplicative character of F p n .Let { ω , . . . , ω n } be an arbitrary basis for F p n over F p . Then the elements of F p n have aunique representation as ξ = x ω + · · · + x n ω n , ≤ x i < p. (4)We denote by B a box in the n -dimensional space, defined by N j < x j ≤ N j + H j , ≤ j ≤ n, (5)where N j , H j are integers satisfying 0 ≤ N j < N j + H j < p for all j .For A ⊂ F q , we denote by E ( A ) := |{ ( x , x , x , x ) ∈ A × A × A × A : x x = x x }| (6)the multiplicative energy of A .As usual, ‘ O ’ and ‘ ≪ ’ denote respectively Landau and Vinogradov symbol, in which theconstants implied depend only on n throughout this paper. Davenport and Lewis [9] proved in 1963 that
Theorem 2.
Let H j = H for ≤ j ≤ n with H > p n n +1) + ε for some ε > , nd let p > p ( ε ) , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈ B χ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ( Hp − δ ) n , where δ = δ ( ε ) > .Remark. We see that if n = 1, the exponent in Theorem 2 is still 1 / ε , which recoversBurgess’ result. While as n increases, the exponent n n +1) will be near to 1 / F p n . Theorem 3.
Let χ be a nontrivial multiplicative character of F p n . Given ε > , there is τ > ε / such that if B = ( n X j =1 x j ω j : x j ∈ ( N j , N j + H j ] ∩ Z , ≤ j ≤ n ) is a box satisfying n Y j =1 H j > p ( + ε ) n then for p > p ( ε ) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈ B χ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ n | B | p − τ , unless n is even and χ | F is principal, where F is the subfield of size p n/ , in which case, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈ B χ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max ξ | B ∩ ξF | + O n ( | B | p − τ ) . Remark.
Theorem 3 also holds if we replace the assumption Q nj =1 H j > p ( + ε ) n by the strongerone H j > p / ε , for all j, which improved upon Davenport and Lewis [9] for n >
4. But, for higher-dimensional gener-alization, the results do not achieve the strength of Burgess [2].We note that Burgess’ strength is obtained only for some special cases, see Burgess [4],Karatsuba [12] and Chang [8]. 4he main ingredient in Chang [7] is the following estimate for the multiplicative energy.
Proposition 4.
Let { ω , . . . , ω n } be a basis for F p n over F p , and let B ⊂ F p n be the box B = ( n X j =1 x j ω j : x j ∈ [ N j + 1 , N j + H j ] , j = 1 , . . . , n ) where ≤ N j < N j + H j < p for all j . Assume that max j H j <
12 ( √ p − . (7) Then we have E ( B, B ) < C n (log p ) | B | / for an absolute constant C < / .Remark. Using a result of Perel’muter and Shparlinski [16] and some sophisticated arguments,Chang removed the influence of the condition (7) on Theorem 3.On the conference of 26th Journ´ees Arithm´etiques (Saint-Etienne, July 2009), using themethod in geometry of numbers (see [1], [17]), Konyagin [13] improved Chang’s estimate formultiplicative energy if H i = H , 1 ≤ i ≤ n . Proposition 5. If H = · · · = H n ≤ p / , then E ( B ) ≪ | B | log p. Then, incorporating the estimate of Proposition 5 into Burgess’ amplification process,Konyagin proved
Theorem 6.
Let χ be a nontrivial multiplicative character of F p n and < ε ≤ / be given.If n ≥ and B is a box defined in (5) and satisfying H j ≥ p / ε , ≤ j ≤ n, then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈ B χ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ | B | p − ε / . .2 Weil’s theorem We will need the following version of Weil’s bound on exponential sums. See [11, Theorem11.23].
Theorem 7 (A. Weil) . Let χ be a nontrivial multiplicative character of F p n of order d > .Suppose f ∈ F p n [ x ] has m distinct roots and f is not a d -th power. Then for n ≥ we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ F pn χ ( f ( x )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( m − p n . The following two theorems generalize Theorem 3 and Theorem 6 respectively.
Theorem 8.
Let χ be a nontrivial multiplicative character of F p n . Given ε > , there is τ > ε / such that if B is a box defined in (5) and satisfying n Y j =1 H j ≥ p ( + ε ) n , then for p > p ( ε ) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈ B χ ( x ) e p (Tr( ax )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ | B | p − τ , unless n is even and χ | F is principal, where F is the subfield of size p n/ , in which case, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈ B χ ( x ) e p (Tr( ax )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max ξ | B ∩ ξF | + O n ( p − τ | B | ) . Theorem 9.
Let χ be a nontrivial multiplicative character of F p n and < ε ≤ / be given.If n ≥ and B is a box defined in (5) and satisfying H j ≥ p / ε , ≤ j ≤ n, then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈ B χ ( x ) e p (Tr( ax )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ | B | p − ε / . Proof of Theorem 8
We incorporate the technique of Chamizo [6] into the argument of Chang [7].
Proof of Theorem 8.
We first prove the theorem under the restriction H j <
12 ( √ p −
1) for all j, (8)which is inherited from the estimate for multiplicative energy in Proposition 4.By breaking up B in smaller boxes, we may assume n Y j =1 H j ∼ p ( + ε ) n . (9)Let δ > I = [1 , p δ ] ,B = ( n X j =1 x j ω j : x j ∈ [0 , p − δ H j ] , j = 1 , . . . , n ) and B I = ( n X j =1 x j ω j : x j ∈ [0 , p/ | I | n ] , j = 1 , . . . , n ) . Since B I ⊂ nP nj =1 x j ω j : x j ∈ [0 , p − δ H j ] , j = 1 , . . . , n o , clearly (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈ B χ ( x ) e p (Tr( ax )) − X x ∈ B χ ( x + yz ) e p (Tr( a ( x + yz ))) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < | B \ ( B + yz ) | + | ( B + yz ) \ B | < np − δ | B | for y ∈ B , z ∈ I . Hence X x ∈ B χ ( x ) e p (Tr( ax ))= 1 | B | | I | X x ∈ B, y ∈ B , z ∈ I χ ( x + yz ) e p (Tr( a ( x + yz ))) + O ( np − δ | B | ) .
7e now estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ B, y ∈ B , z ∈ I χ ( x + yz ) e p (Tr( a ( x + yz ))) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X x ∈ B, y ∈ B (cid:12)(cid:12)(cid:12)(cid:12) X z ∈ I χ ( x + yz ) e p (Tr( a ( x + yz ))) (cid:12)(cid:12)(cid:12)(cid:12) ≤ X u ∈ F q ν ( u ) sup y ∈ B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X z ∈ I χ ( u + z ) e p (Tr( ay ( u + z ))) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X u ∈ F q ν ( u ) sup b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X z ∈ I χ ( u + z ) e p (Tr( b ( u + z ))) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where ν ( u ) = (cid:12)(cid:12)(cid:12)n ( x, y ) ∈ B × B : xy = u o(cid:12)(cid:12)(cid:12) . Then X x ∈ B χ ( x ) e p (Tr( ax )) ≤ | B | | I | X u ∈ F q ν ( u ) sup b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X z ∈ I χ ( u + z ) e p (Tr( b ( u + z ))) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ( np − δ | B | ) ≤ | B | | I | X u ∈ F q ν ( u ) sup b | I | q X cc − b ∈ B I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X z ∈ I χ ( u + z ) e p (Tr( c ( u + z ))) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ( np − δ | B | ) ≤ | B | q X u ∈ F q ν ( u ) X cc − b ∈ B I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X z ∈ I χ ( u + z ) e p (Tr( c ( u + z ))) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ( np − δ | B | )with the sum X cc − b ∈ B I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X z ∈ I χ ( u + z ) e p (Tr( c ( u + z ))) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) attains its maximum at b ∈ F q .Let r ≥ X u ∈ F q ν ( u ) X cc − b ∈ B I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X z ∈ I χ ( u + z ) e p (Tr( c ( u + z ))) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ V − r V r W r , V = X u ∈ F q ν ( u ) , V = X u ∈ F q ν ( u ) ,W = X u ∈ F q X cc − b ∈ B I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X z ∈ I χ ( u + z ) e p (Tr( c ( u + z ))) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r . Observe that V = | B || B | and V = |{ ( x , x , y , y ) ∈ B × B × B × B : x y = x y }| = X v |{ ( x , x ) : x x = v }| |{ ( y , y ) : y y = v }|≤ E ( B, B ) E ( B , B ) < n +1 (log p ) | B | | B | < n +1 (log p ) | B | p − nδ , by the Cauchy-Schwarz inequality, Proposition 4 and the definition of B .Now we bound W . Recall that q = p n . Then X u ∈ F q X cc − b ∈ B I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X z ∈ I χ ( u + z ) e p (Tr( c ( u + z ))) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r ≤ ( q/ | I | ) r − X u ∈ F q X c ∈ F q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X z ∈ I χ ( u + z ) e p (Tr( c ( u + z ))) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r ≤ ( q/ | I | ) r − X z , ..., z r ∈ I (cid:12)(cid:12)(cid:12)(cid:12) X u ∈ F q χ (( u + z ) · · · ( u + z r )( u + z r +1 ) q − · · · ( u + z r ) q − ) × X c ∈ F q e p (Tr( c ( z + · · · + z r − z r +1 − · · · − z r ))) (cid:12)(cid:12)(cid:12)(cid:12) = q r | I | − r X z , ..., z r ∈ Iz + ··· + z r = z r +1 + ··· + z r (cid:12)(cid:12)(cid:12)(cid:12) X u ∈ F q χ (( u + z ) · · · ( u + z r )( u + z r +1 ) q − · · · ( u + z r ) q − ) (cid:12)(cid:12)(cid:12)(cid:12) . z , . . . , z r ∈ I such that at least one of the elements is not repeated twice, thepolynomial f z ,...,z r ( u ) = ( u + z ) · · · ( u + z r )( u + z r +1 ) q − · · · ( u + z r ) q − clearly cannot be a d -th power. Since f z ,...,z r ( u ) has no more than 2 r many distinct roots, Theorem 7 gives (cid:12)(cid:12)(cid:12)(cid:12) X u ∈ F q χ (( u + z ) · · · ( u + z r )( u + z r +1 ) q − · · · ( u + z r ) q − ) (cid:12)(cid:12)(cid:12)(cid:12) < rp n . For those z , . . . , z r ∈ I such that every root of f z ,...,z r ( u ) appears at least twice, webound P(cid:12)(cid:12)P u ∈ F q χ ( f z ,...,z r ( u )) (cid:12)(cid:12) by q times the number of such z , . . . , z r . Since there areat most r roots in I and for each z , . . . , z r there are at most r choices, we obtain a bound | I | r r r p n .Therefore X u ∈ F q X c ∈ F q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X z ∈ I χ ( u + z ) e p (Tr( c ( u + z ))) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r ≤ q r | I | ( r r | I | − r p n + 2 rp n )and W r ≤ q | I | r ( r | I | − p n r + 2 p n r ) . Putting the above estimates together, we have1 | B | q V < nr (log p )( | B || B | ) − r | B | r p − nr δ | I | r (cid:16) r | I | − p n r + 2 p n r (cid:17) < nr (log p ) p nr δ − nr δ | B | − r | I | r (cid:16) r | I | − p n r + 2 p n r (cid:17) < r · nr | B | − r p n r + nr δ + δ r log p< r · nr | B | p n r + nr δ − nr ( + ε )+ δ r log p< r · nr | B | p − nr ( ε − δ )+ δ r log p. The second-to-last inequality holds because of (9) and by assuming δ ≥ n r .Similar to the argument of Chang [7], we can show that p − nr ( ε − δ )+ δ r log p < p − ε / . Then we prove the theorem under the condition (8).Now we are at the position to remove the additional hypothesis (8) on the shape of B .We proceed in several steps and rely essentially on a further key ingredient provided by thefollowing estimate in Perel’muter and Shparlinski [16].10 roposition 10. Let χ be a nonprincipal multiplicative character of F q and let g ∈ F q be agenerating element, i.e. F q = F p ( g ) . Then for any a ∈ F p , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X t ∈ F p χ ( g + t ) e p ( at ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ np / . (10)First we make the following observation.Let H ≥ H ≥ . . . ≥ H n . If H < p + ε , we may clearly write B as a disjoint unionof boxes B α ⊂ B satisfying the first condition in (8) and | B α | > ( p − ε ) n | B | > − n p ( + ε ) n .Since (8) holds for each B α , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ B α χ ( x ) e p (Tr( ax )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < cnp − τ | B α | . Hence (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈ B χ ( x ) e p (Tr( ax )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < cnp − τ | B | . Therefore we may assume that H > p + ε . Case 1. n is odd. We denote I i = [ N i + 1 , N i + H i ]. Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈ B χ ( x ) e p (Tr( ax )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x i ∈ I i ≤ i ≤ n X x ∈ I χ (cid:18) x + x ω ω + · · · + x n ω n ω (cid:19) e p (Tr( ax ω + x ω + · · · + x n ω n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( x ,...,x n ) ∈ D c X x ∈ I χ (cid:18) x + x ω ω + · · · + x n ω n ω (cid:19) e p (Tr( ax ω + x ω + · · · + x n ω n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( x ,...,x n ) ∈ D X x ∈ I χ (cid:18) x + x ω ω + · · · + x n ω n ω (cid:19) e p (Tr( ax ω + x ω + · · · + x n ω n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) with D = (cid:26) ( x , . . . , x n ) ∈ I × · · · × I n : F p (cid:18) x ω ω + · · · + x n ω n ω (cid:19) = F q (cid:27) D c = I × · · · × I n \ D. Using (10) we estimate the first sum as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( x ,...,x n ) ∈ D c X x ∈ I χ (cid:18) x + x ω ω + · · · + x n ω n ω (cid:19) e p (Tr( ax ω + x ω + · · · + x n ω n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x i ∈ I i ≤ i ≤ n e p (Tr( a ( x ω + · · · + x n ω n ))) X x ∈ I χ (cid:18) x + x ω ω + · · · + x n ω n ω (cid:19) e p (Tr( ax ω )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X x i ∈ I i ≤ i ≤ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ I χ (cid:18) x + x ω ω + · · · + x n ω n ω (cid:19) e p (Tr( aω ) x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = X x i ∈ I i ≤ i ≤ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ F p χ (cid:18) x + x ω ω + · · · + x n ω n ω (cid:19) e p (Tr( aω ) x ) · p X b ∈ F p X x ′ ∈ I e p ( b ( x − x ′ )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ p X x i ∈ I i ≤ i ≤ n X b ∈ F p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ F p χ (cid:18) x + x ω ω + · · · + x n ω n ω (cid:19) e p ((Tr( aω ) + b ) x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ′ ∈ I e p ( − bx ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ p np / | B | H X b ∈ F p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ′ ∈ I e p ( bx ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( n ) p log p | B | H . For the second sum, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( x ,...,x n ) ∈ D X x ∈ I χ (cid:18) x + x ω ω + · · · + x n ω n ω (cid:19) e p (Tr( ax ω + x ω + · · · + x n ω n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ p | D | ≤ p X G (cid:12)(cid:12)(cid:12)(cid:12) G \ Span F p (cid:18) ω ω , . . . , ω n ω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , where G runs over nontrivial subfields of F q . Since q = p n and n is odd, obviously [ F q : G ] ≥
3. Hence [ G : F p ] ≤ n . Furthermore, since { ω , . . . , ω n } is a basis of F q over F p ,1 / ∈ Span F p ( ω ω , . . . , ω n ω ) and the proceeding implies thatdim F p (cid:18) G \ Span F p (cid:18) ω ω , . . . , ω n ω (cid:19)(cid:19) ≤ n − . | H | , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈ B χ ( x ) e p (Tr( ax )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( n )((log p ) p − ε | B | + p n ) < (cid:0) c ( n )(log p ) p − ε + p − n (cid:1) | B | , since | B | > p n . This proves our claim. Case 2. n is even. In view of the earlier discussion, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈ B χ ( x ) e p (Tr( ax )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( x ,...,x n ) ∈ D c X x ∈ I χ (cid:18) x + x ω ω + · · · + x n ω n ω (cid:19) e p (Tr( ax ω + x ω + · · · + x n ω n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( x ,...,x n ) ∈ D X x ∈ I χ (cid:18) x + x ω ω + · · · + x n ω n ω (cid:19) e p (Tr( ax ω + x ω + · · · + x n ω n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where D = (cid:26) ( x , . . . , x n ) ∈ I × · · · × I n : (cid:18) x ω ω + · · · + x n ω n ω (cid:19) ∈ F (cid:27) ,D c = I × · · · × I n \ D , and F is the subfield of size p n/ .Our only concern is to bound the second sum, namely ̟ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( x ,...,x n ) ∈ D X x ∈ I χ (cid:18) x + x ω ω + · · · + x n ω n ω (cid:19) e p (Tr( ax ω + x ω + · · · + x n ω n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . First, we note that since 1 , ω ω , . . . , ω n ω are independent, ω j ω ∈ F for at most n − j ’s.After reordering, we may assume that ω j ω ∈ F for 2 ≤ j ≤ k and ω j ω / ∈ F for k + 1 ≤ j ≤ n ,where k ≤ n . we also assume that H k +1 ≤ · · · ≤ H n . Fix x , . . . , x n − . Obviously there is nomore than one value of x n such that x ω ω + · · · + x n ω n ω ∈ F , since otherwise ( x n − x ′ n ) ω n ω ∈ F with x n = x ′ n contradicting the fact that ω n ω / ∈ F .Therefore, | D | ≤ | I | · · · | I n − | ̟ ≤ | B | H n . If H n > p τ , we are done. Otherwise H k +1 · · · H n ≤ p ( n − k ) τ < p τ . (11)Define B = (cid:26) x + x ω ω + · · · + x k ω k ω : x i ∈ I i , ≤ i ≤ k (cid:27) . (12)Hence B ⊂ F and by (11) | B | > | B | H k +1 · · · H n > p − τ n > p n . (We assume τ < .)Clearly, if ( x , . . . , x n ) ∈ D , then z = x k +1 ω k +1 ω + · · · + x n ω n ω ∈ F . Assume χ | F non-principal. Then by completing the sum over y and recalling the classical estimates for Gaussiansums in finite fields [14, Theorem 5.11] and (12), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X y ∈ B χ ( y + z ) e p (Tr( aω ( y + z ))) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (log p ) n max ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ F ψ ( x ) χ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (log p ) n | F | ≤ p − n | B | , where ψ runs over all additive characters. Therefore, clearly ̟ ≤ H k +1 · · · H n p − n | B | = p − n | B | providing the required estimate.If χ | F is principal, then obviously ̟ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ I X ( x ,...,x n ) ∈ D e p (Tr( a ( x ω + · · · + x n ω n ))) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ H | D | = (cid:12)(cid:12)(cid:12)(cid:12) F ∩ ω B (cid:12)(cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈ B χ ( x ) e p (Tr( ax )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | ω F ∩ B | + O n ( p − τ | B | ) . This completes the proof of Theorem 8. 14
Proof of Theorem 9
Similar to the proof of Theorem 8, by breaking up B in smaller boxes, we may assume H ≍ · · · ≍ H n ≍ p + ε . Then, using the arguments in the proof of Theorem 8, we can prove Theorem 9 along thelines of Konyagin [13].
Acknowledgements
The author thanks Professor Igor Shparlinski for his helpful comments on an earlier versionof the paper. The author also thanks Professor Sergei Konyagin for sending his preprint.Part of this work was done while the author was visiting the Morningside Center of Math-ematics, whose hospitality is gratefully acknowledged. The author was supported by theNational Natural Science Foundation of China (Grant No. 10671056).
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