EXPLICIT BOUNDS ON MONOMIAL AND BINOMIAL EXPONENTIAL SUMS

EXPLICIT BOUNDS ON MONOMIAL AND BINOMIAL EXPONENTIAL SUMS TODD COCHRANE AND CHRISTOPHER PINNER Abstract. Let p be a prime and ep (·) = e2πi·/p . First, we make explicit the monomial sum bounds of Heath-Brown and Konyagin: P p−1 5/8 p5/8 , λ d3/8 p3/4 }, d x=1 ep (ax ) ≤ min{λ d √ 4 where λ = 2/ 3 = 1.51967.... Second, letting d = (k, l, p − 1), we obtain the explicit binomial sum bound P p−1 13/46 p89/92 , k l x=1 ep (ax + bx ) ≤ (k − l, p − 1) + 2.292 d for any nonconstant binomial axk + bxl on Zp , by sharpening the estimate for the number of solutions of the system xk1 + xk2 = xk3 + xk4 , xl1 + xl2 = xl3 +xl4 . Finally, we apply the latter estimate to establish the Goresky-Klapper conjecture on the decimation of ℓ-sequences for p > 4.92 · 1034 . 1. Introduction For prime p and polynomial f (x) over Z, let S(f ) denote the exponential sum, S(f ) = p−1 X ep (f (x)), x=1 where ep (·) = e2πi·/p is the additive character on Zp . The need for precise numeric estimates for such sums has become apparent in many areas of mathematics. For instance, to quantify the distribution of k-th powers (mod p) one needs estimates for the monomial sum S(xk ) and the binomial sum S(xk +bx). Such estimates were used by Bourgain, Paulhus and the authors [7], to resolve the Goresky-Klapper conjecture on the decimation of l-sequences for all sufficiently large primes, a problem of interest to computer scientists; see Section 7. In that paper we were able to establish the validity of the conjecture for p > 2.26 · 1055 . A computer has verified the conjecture for p < 2 · 106 . In order to close this gap it is useful to have more precise estimates for a binomial sum. In this paper we obtain numeric estimates for S(f ) in the cases where f is a monomial or binomial. In particular the bounds obtained allow us to establish the Goresky-Klapper conjecture for p > 4.92 · 1034 . First we make explicit the monomial exponential sum bounds of Heath-Brown and Konyagin [14]. Theorem 1.1. For any prime p, multiplicative subgroup A of Z∗p and integer a √ with p ∤ a, we have, with λ = 2/ 4 3 = 1.51967...,  1/2  |A| > 13 p2/3 p , X ep (ax) ≤ λ |A|3/8 p1/4 , p1/2 < |A| ≤ 13 p2/3   5/8 1/8 x∈A λ |A| p , 3 p1/3 < |A| ≤ p1/2 . Date: October 5, 2009. 1 2 TODD COCHRANE AND CHRISTOPHER PINNER Equivalently, letting A ⊂ Z∗p be the subgroup of d-th powers we have Theorem 1.2. Let p be a prime and d a positive integer with d|(p − 1). Then for any integer a with p ∤ a,  1/2  d < 3 p1/3 , p−1 (d − 1)p + 1, X ep (axd ) ≤ λ d5/8 p5/8 , 3 p1/3 ≤ d < p1/2 ,   x=1 λ d3/8 p3/4 , p1/2 ≤ d < 31 p2/3 , with λ = 2 √ 4 3 = 1.51967... . Each of the bounds in Theorems 1.1 and 1.2 is valid for arbitrary d. We have indicated to the right the interval where the bound is optimal. The first bound in each of these theorems is just the classical bound for a Gauss sum. For |A| < 3p1/3 , or equivalently d > 13 p2/3 all of these bounds are trivial. Konyagin [15] has obtained 1 nontrivial bounds for |A| > p 4 +ǫ that can be made explicit, but he (and we) have P not computed the constants. Bourgain and Garaev [4] also have the bound | x∈A ep (ax)| ≪ |A|.999981.. for any A with |A| > p1/4 , but the implied constant has not been computed. Bourgain and Konyagin [6], and Bourgain, Glibichuck and Konyagin [5] obtained estimates valid for |A| > pǫ . More recently Bourgain [3] has proved that X log p ep (ax) < p− exp(−C log |A| ) |A| x∈A for√some absolute (undetermined) constant C > 1. For example, to save a factor e− log p on the trivial bound one needs only log |A| > 2C log p/ log log p. Next, we turn to binomial sums. Let a, b, k, l be integers and f (x) = axk + bxl . We shall only insist that f be nonconstant on Zp . Thus, it is allowed to collapse to a nonconstant monomial. Set d = (k, l, p − 1). In [8, Corollary 1.1] the authors established the upper bound p−1 X x=1 ep (axk + bxl ) ≪ (k − l, p − 1) + d3/13 p51/52 , for any nonconstant binomial f (x) on Zp , a bound that is nontrivial for d ≪ p1/12 . Here we establish a stronger (and explicit) bound, that remains nontrivial for d ≪ p3/26 . Theorem 1.3. For any nonconstant binomial f (x) on Zp we have p−1 X x=1 ep (axk + bxl ) ≤ (k − l, p − 1) + 2.292 d13/46 p89/92 . The first term can be removed if −ba is not a (k − l)-th power in Zp . The term d∗ := (k − l, p − 1) cannot be removed from the right-hand side when −bā is a d∗ -th power. Indeed, in this case we see that S(f ) ∼ d∗ if d∗ ≫ min{d1/2 p3/4 , d5/13 p10/13 } by estimate (29) and the bounds in Theorem 1.2. In Theorem 4.1 we give a slightly stronger upper bound, nontrivial for d < p1/3 but requiring (k, p − 1) or (l, p − 1) to be sufficiently large. By the work of Bourgain [2], BINOMIAL SUMS 3 it is known that if d < p1−ǫ and d∗ < p1−ǫ , then |S(f )| ≤ p1−δ for some δ = δ(ǫ); see [11]. Crucial for our binomial bounds will be estimates for M (k, l), the number of solutions in (Z∗p )4 of the system of simultaneous equations xk1 + xk2 = xk3 + xk4 xl1 + xl2 = xl3 + xl4 . For example when the exponents 1 ≤ l < k have (kl, p − 1) = 1 and (k ∓ l, p − 1) ≤ 16/23 1.68 (p − 1) we obtain the bound (see Theorem 7.1) M (k, ±l) ≤ 27.57p66/23 . The special case l = 1 of this will be used in Corollary 7.1 when verifying the Goresky-Klapper conjecture, 27.57 replacing the 13658 in Theorem 3 of our earlier paper [7]. Good bounds for M (k, l) translate immediately into good bounds for the corresponding binomial exponential sum via p−1 X x=1 ep (axk + bx±l ) ≤ M (k, ±l)1/4 p1/4 . We should remark that the various Pp−1inequalities above in fact hold for the more general mixed exponential sum x=1 χ(x)ep (f (x)), where χ is a multiplicative character mod p and f is a monomial or binomial. 2. Proofs of Theorems 1.1 and 1.2 Define N (A) = |{(x1 , x2 , y1 , y2 ) ∈ A4 : x1 + x2 = y1 + y2 }|, and for any a ∈ Fp let N (A, a) = |{(x1 , x2 ) ∈ A2 : x1 + x2 = a}|. In order to pass from the estimate of N (A) to the estimate of the monomial exponential sum, we use Lemma 2.1. [14] For any subgroup A of Z∗p , ( 1 1 1 X N (A) 4 |A|− 4 p 4 , ep (ax) ≤ 1 1 N (A) 4 p 8 . x∈A 2/3 If |A| then Theorem 1.1 follows from the classical estimate for a Gauss P≥ p √ sum, | x∈A ep (ax)| ≤ p. For |A| < p2/3 it is an immediate consequence of Lemma 2.1 and Theorem 2.1. For any multiplicative subgroup A of Zp with |A| < p2/3 we have 5/2 . N (A) ≤ 16 3 |A| The classical estimate of Hua, Vandiver and Weil for the number of solutions of the homogeneous equation xd1 + xd2 = xd3 + xd4 can be stated in the manner 4 4 2/3 5/2 one has |N (A)| < |A| . One also |N (A) − |A| p | < p|A|. Thus for |A| ≥ p p + |A| 2/3 sees that the hypothesis |A| < p in the theorem cannot be relaxed. To obtain the constant 16/3 in the theorem, we make use of the following lemma of Mattarei [17], for counting the number of solutions of the Fermat equation xd + y d = z d over 4 TODD COCHRANE AND CHRISTOPHER PINNER a finite field. It is a refinement of a result of Garcia and Voloch [12]. A similar upper bound is also given in [16] with an undetermined constant. Lemma 2.2. For any nonzero a ∈ Zp and multiplicative subgroup A of Zp with |A| < (1/4)1/4 (p − 1)3/4 , we have N (A, a) ≤ 3 · 2−2/3 |A|2/3 . The result of [17] has an extra hypothesis that d ≥ 4, where d = (p − 1)/|A|, but one can check that the lemma holds trivially for d < 4. Indeed, for d = 3, N (A, a) ≤ |A| ≤ 3 · 2−2/3 |A|2/3 provided that |A| ≤ 6. If |A| ≥ 7 then since p = 3|A| + 1 ≥ 22 we have |A| = (p − 1)/3 ≥ (1/4)1/4 (p − 1)3/4 , contrary to assumption. A similar argument applies when d = 1 or 2. In [12] the upper bound 4|A|2/3 is obtained for |A| < (p − 1)/((p − 1)1/4 + 1). Let A be a multiplicative subgroup of Zp , with t = |A|. We start by writing A + A as a disjoint union of cosets of A, A + A = Ax1 ∪ Ax2 · · · ∪ Axn ∪ {0}, where {0} is omitted if −1 ∈ / A. For any coset Axj let Nj = |{x ∈ A : x + 1 ∈ Axj }| = |{(x, y) ∈ A × A : x + y = xj }|. We assume the sets Axi have been ordered so that N1 ≥ N2 ≥ N3 ≥ · · · ≥ Nn . Now for any x ∈ A, x 6= −1, x + 1 ∈ Axj for some j and so n X (1) j=1 where Nj = t − δ, ( 1, δ= 0, and if −1 ∈ A, if −1 ∈ / A, N (A) = δt2 + t (2) n X Nj2 . j=1 The next lemma is extracted from the proof of [16, Lemma 3.2]. Lemma 2.3. Let a, b, d, s be positive integers such that s ≤ n, sad + 12 sd(d − 1) < ab2 , ab ≤ t, tb < p, where t = |A|. Then s X a − 1 + 2t(b − 1) Nj ≤ . d j=1 Proof. The lower case a, b, d in the lemma correspond to the upper case A, B, D in [16]. In equation (3.11) of [16] one actually has sad + 12 sd(d − 1) < ab2 by summing over k in the preceding line of their proof.  Apply the lemma with b = [(4st)1/3 ] + 1, Then a = [t/b], d = 2a. 4t 1 sad + sd(d − 1) < 4a2 s = 4a(as) ≤ (as) ≤ ab2 , 2 b BINOMIAL SUMS 5 and so if tb ≤ p then we deduce s X 1 − 1b 1 t(b − 1) 1 t(b − 1) 1 1 + ≤ + t = + b2 Nj ≤ − (3) . 2 2a a 2 2 1 − bt b −1 j=1 If we assume further that b2 < t we get from (3), s X 1 N j ≤ + b2 . (4) 2 j=1 If b2 ≥ t then the same bound holds trivially by (1). Since the left-hand side is an integer the 12 can be dropped, thus establishing Lemma 2.4. For any positive integer s ≤ n such that bt < p, s X Nj ≤ ([(4ts)1/3 ] + 1)2 . j=1 5 and 8 will require us to asymptotically evaluate sums of the form P Sections −c j . Hence for 0 < c < 1 we define j≤s Z s c {x}x−1−c dx. +c (5) γc (s) = 1−c 1 In §5 we will need estimates for the quantity   214/3 16 (6) κ0 (s) = − γ1/3 (s), γ2/3 (s) + 25/3 + 9 9 and in §8 8 (7) κ1 (s) = γ2/5 (s) − γ4/5 (s). 3 Lemma 2.5. For 0 < c < 1 and s in N X s1−c − γc (s). (8) j −c = 1−c j≤s The functions κ0 (s) and κ1 (s) are increasing for s in N with (9) and (10) for all s in N, with κ0 (s) < −2.083, κ1 (s) < −1.4 20 . 9 Proof. Partial summation gives (8). Claims (9) and (10) follow from Z  25/3 ∞ {x}  7/3 1/3 (9 + 2 )x − 16 dx, κ0 (s) = κ0 (∞) − 27 s x5/3   Z ∞ 4 {x} 4 2/5 x − 1 dx, κ1 (s) = κ1 (∞) − 5 s x9/5 3 (11) κ1 (1) = − (checking numerically that κ0 (1) < κ0 (2)) and numerical computation κ0 (∞) < −2.083 and κ1 (∞) < −1.4.  6 TODD COCHRANE AND CHRISTOPHER PINNER 3. Proof of Theorem 2.1 5/2 Suppose t < p2/3 . Since N (A) ≤ t3 ≤ 16 for t ≤ 28 we may assume that 3 t 2/3 3/4 t ≥ 29 and p ≥ 157. Hence t < p < .7(p − 1) , and by (2) and (1) and Lemma 2.2 we have N (A) ≤ δt2 + tN1 n X j=1 Nj = δt2 + tN1 (t − δ) ≤ t2 + 3 · 2−2/3 t5/3 (t − 1) < 16 5/2 t 3 for t ≤ 485. Hence we assume that t ≥ 486 and, setting h√ i t/4 , J= that J ≥ 5. We define mj = X 27/3 2/3 −1/3 t j , wj = Nj − mj , C(s) = wj . 3 j≤s From Lemma 2.2 C(s) = X j≤s Nj − X j≤s giving  7/3 2 mj ≤ t2/3 3 · 2−2/3 s − 3 X j≤s  j −1/3  , C(1) ≤ 0.210t2/3 , C(2) ≤ 0.767t2/3 , C(3) ≤ 1.492t2/3 . (12) For s ≤ J − 1 ≤ 41 t1/2 − 1 we have 2 b ≤ 1/3 1/3 4 t  1 1/2 t −1 4 1/3 !2 +1 < t,  
2 P bt < t3/2 < p, and Lemma 2.4 gives j≤s Nj ≤ 22/3 t1/3 s1/3 + 1 . Hence, using the notation (5) and Lemma 2.5, for s ≤ J − 1 i 2 27/3 X t2/3 j −1/3 22/3 t1/3 s1/3 + 1 − 3 j≤s   7/3 2 3 ≤ (22/3 t1/3 s1/3 + 1)2 − s2/3 − γ1/3 (s) t2/3 3 2 C(s) ≤ (13) = h 27/3 27/3 γ1/3 (s)t2/3 + 25/3 t1/3 s1/3 + 1 ≤ γ1/3 (J − 1)t2/3 + 2t1/2 , 3 3 and by Lemma 2.4 (14) −1 Ns ≤ s s X i=1 Ni ≤  
2 22/3 t1/3 s1/3 + 1 . s BINOMIAL SUMS 7 Using (1) and (2) we write n X N (A)t−1 = Nj2 + δt j=1 X ≤ Nj2 + NJ j<J X = j<J ≤ Nj + δt j≥J Nj (Nj − NJ ) + NJ (t − δ) + δt X mj (Nj − NJ ) + X X j<J = X m2j + j<J j<J X j<J wj (Nj − NJ ) + NJ (t − 1) + t wj (Nj − NJ ) + = M1 + E1 + E2 + E3 + t, X j<J  mj wj + NJ t − 1 − X j<J  mj  + t where M1 = X m2j = j<J j<J 14/3 = 214/3 4/3 X −2/3 t j 9 2 9   t4/3 3(J − 1)1/3 − γ2/3 (J − 1) 214/3 4/3 16 3/2 214/3 t − γ2/3 (J − 1)t4/3 − t = 3 9 3 !  √ 1/3 t − (J − 1)1/3 4 and  E3 = NJ t − 1 − = NJ  = NJ X j<J  mj  27/3 2/3 t 3   3 (J − 1)2/3 − γ1/3 (J − 1) 2 ! !  √ 2/3 27/3 t 2/3 2/3 4/3 2/3 − (J − 1) −1 . γ1/3 (J − 1)t + 2 t 3 4 t−1− Using partial summation (eg Hardy & Wright Theorem 421), (13) and N1 ≤ 3 · 2−2/3 t2/3 , X X wj (Nj − NJ ) = C(J − 1)(NJ−1 − NJ ) + E1 = C(j)(Nj − Nj+1 ) j<J ≤  1≤j≤J−2 7/3 2 3 γ1/3 (J − 1)t2/3 + 2t1/2  X 1≤j≤J−1 (Nj − Nj+1 )  27/3 γ1/3 (J − 1)t2/3 + 2t1/2 (N1 − NJ ) 3   7/3 2 2/3 1/2 5/3 4/3 1/3 7/6 γ1/3 (J − 1)t + 2t . ≤ 2 γ1/3 (J − 1)t + 3 · 2 t − NJ 3 =  8 TODD COCHRANE AND CHRISTOPHER PINNER Similarly, using the bounds (12) on C(j) for j ≤ 3 and (13) for j ≥ 4, X 27/3 2/3 X mj wj = wj j −1/3 E2 = t 3 j<J j<J     X 27/3 2/3  = t C(J − 1)(J − 1)−1/3 + C(j) j −1/3 − (j + 1)−1/3  3 1≤j≤J−2 7/3 2  t4/3 0.210(1 − 2−1/3 ) + 0.767(2−1/3 − 3−1/3 ) + 1.492(3−1/3 − 4−1/3 ) 3    7/3  7/3 X 2 2 + t2/3 γ1/3 (J − 1)t2/3 + 2t1/2 (J − 1)−1/3 + (j −1/3 − (j + 1)−1/3 ) 3 3 4≤j≤J−2   8/3 16 2 ≤ γ1/3 (J − 1) + 0.361 t4/3 + t7/6 . 9 3 ≤  Hence 16 3/2 t + t4/3 (E4 + E5 ) 3 where, with κ0 (J − 1) as defined in (6), N (A)t−1 ≤ E4 = κ0 (J − 1) + 0.361 + 5.897t−1/6 + t−1/3 , and E5 = N J 4/3 2 ! !  √ 2/3 214/3 t 2/3 −2/3 −5/6 −4/3 − (J − 1) t − 2t −t − 4 3 !  √ 1/3 t 1/3 − (J − 1) . 4 Also, from (14), NJ ≤ NJ−1 ≤  
2 22/3 (J − 1)1/3 t1/3 + 1 . (J − 1) For t < 2704 one checks numerically that E4 + E5 < −.4743, whence N (A)t−1 ≤ 16 3/2 . 3 t For t ≥ 2704 we have J ≥ 13. The bounds NJ ≤ (15) (22/3 + 12−1/3 2704−1/3 )2 2/3 (22/3 (J − 1)1/3 t1/3 + 1)2 2.621t2/3 ≤ t ≤ , (J − 1) (J − 1)1/3 (J − 1)1/3  √ 1/3 t 2 − (J − 1)1/3 ≤ (J − 1)−2/3 , 4 3 and, using (15),  √ 2/3 t − (J − 1)2/3 = 4 !  √ 1/3 !  √ 1/3 t t 1/3 (J − 1) + − (J − 1) 4 4 !   √ 1/3  t 2 1/3 −2/3 1/3 − (J − 1) ≤ 2(J − 1) + (J − 1) 3 4 !  √ 1/3 t 1/3 ≤ 2.056 − (J − 1) (J − 1)1/3 , 4 1/3 BINOMIAL SUMS 9 give !   √ 1/3 t 1/3 E5 ≤ 2.621 · 2 − (J − 1) − 2.621 · 2t−1/6 (J − 1)−1/3 4  √ 2/3  √ 1/3 ! t t −2/3 −1/6 −1/3 ≤ 3.409(J − 1) − 5.242t (J − 1) = 3.409 − 5.242 t−1/3 J −1 J −1 2/3 1/3 !   8 8 t−1/3 − 5.242 4 + ≤ 3.409 4 + J −1 J −1  2/3  1/3 ! 14 14 ≤ 3.409 t−1/3 < 0.760t−1/3 . − 5.242 3 3  4/3 214/3 · 2.056 − 3 From Lemma 2.5 we have κ0 (J − 1) < −2.083. Hence for t ≥ 2704 we have 3/2 . E4 + E5 ≤ −1.722 + 5.897t−1/6 + 1.760t−1/3 < 0, and N (A)t−1 < 16 3 t 4. Another Binomial Sum Bound The following theorem is needed in the proof of Theorem 1.3, but it has independent interest. It yields a nontrivial bound on any binomial exponential sum with d ≪ p1/3 and either (k, p − 1) > d or (l, p − 1) > d, where d = (k, l, p − 1). Theorem 4.1. For any nonconstant binomial f (x) = axk + bxl , and constant λ as in Theorem 1.2, we have the bound  1/2 d |S(f )| ≤ p + min{λ8/11 d15/88 p21/22 , λ2/3 d1/8 p23/24 }. (k, p − 1) The proof uses averaging methods similar to what is found in Akulinicev [1], Yu [20] and the author’s work [10], together with the bounds for a monomial sum given in Theorem 1.2. For any integer k, set (16) Φ(k) = max a6=0 p−1 X ep (axk ) . x=1 Lemma 4.1. For any binomial f (x) = axk + bxl , we have |S(f )| ≤ Φ In particular, with λ as in Theorem 1.2, (17) |S(f )| ≤ (18) |S(f )| ≤ (19) |S(f )| ≤ p3/2 d , (l, p − 1) (l, p − 1) > λ p5/4 d5/8 , (l, p − 1)5/8 λ p9/8 d3/8 , (l, p − 1)3/8 √  d(p−1) (l,p−1)  . 1 2/3 dp , 3 pd < (l, p − 1) < 1 2/3 dp , 3 √ (l, p − 1) < d p. The inequality in (18) is a generalization of Yu [20, Theorem 2]. His theorem required l|p−1 and d = 1. From this he deduced the uniform bound |S(f )| ≪ p23/24 under the same constraints. 10 TODD COCHRANE AND CHRISTOPHER PINNER Proof. Set m = p−1 (l,p−1) . Then (p − 1)S(f ) = p−1 p−1 X X ep (f (xy m )) = p−1 p−1 X X ep (axk y km + bxl ) x=1 y=1 y=1 x=1 and so, |S(f )| ≤ p−1 p−1 1 X X ep (axk y km ) . p − 1 x=1 y=1 p−1 The first result follows from the observation that (km, p − 1) = (l,p−1) d. The remaining inequalities are an immediate consequence of Theorem 1.2.  In [10, Lemma 3.1] the authors proved |S(f )| ≤ p  d (k, p − 1) 1/2 + √ p Φ((l, p − 1))1/2 . We deduce from Theorem 1.2, Lemma 4.2. For any nonconstant binomial f (x) = axk + bxl on Zp , (20) |S(f )| ≤ p (21) |S(f )| ≤ p (22) |S(f )| ≤ p  d (k,p−1)  d (k,p−1)  d (k,p−1) 1/2 + p3/4 (l, p − 1)1/2 , 1/2 + λ1/2 p13/16 (l, p − 1)5/16 , 1/2 + λ1/2 p7/8 (l, p − 1)3/16 , (l, p − 1) < 3p1/3 , 3p1/3 ≤ (l, p − 1) < p1/2 , p1/2 ≤ (l, p − 1) < 1 2/3 p , 3 with λ as in Theorem 1.2. Proof of Theorem 4.1. We treat a number of separate cases which may be of independent interest. The theorem itself just needs the argument presented in cases (iv) and (v). (i). If (l, p − 1) > 31 dp2/3 then by (17), |S(f )| < 3p5/6 . √ (ii). If d p < (l, p − 1) ≤ 13 dp2/3 then by (18), |S(f )| < λp15/16 . √ (iii). If (l, p − 1) < 3p1/3 then by (20), |S(f )| < A + p3/4 (3p1/3 )1/2 < A + 3p11/12 , where A is the first term in the theorem. √ (iv). Suppose next that 3p1/3 ≤ (l, p − 1) ≤ p. If (l, p − 1) ≥ λ8/11 p5/11 d6/11 we use (19) to get |S(f )| ≤ λ8/11 d15/88 p21/22 . If (l, p − 1) ≤ λ8/11 p5/11 d6/11 then we use (21) to get the same bound with A added. √ (v). Suppose that p ≤ (l, p − 1) ≤ 31 p2/3 . If (l, p − 1) > λ8/9 d2/3 p4/9 then use (19) to get |S(f )| ≤ λ2/3 d1/8 p23/24 . If (l, p − 1) ≤ λ8/9 d2/3 p4/9 , then we use (22) to get the same with A added.  BINOMIAL SUMS 11 5. Lemmas for Theorem 1.3 For any integers k, l let M (k, l) denote the number of solutions in (Z∗p )4 of the system xk1 + xk2 = xk3 + xk4 xl1 + xl2 = xl3 + xl4 , and put M+ (k, l) = M (k, l) for 1 ≤ l < k < p − 1, M− (k, l) = M (k, −l) for 1 ≤ l < k, k + l < p − 1. Let (23) S+ (k, l) = p−1 X x=1 and (24) S− (k, l) = p−1 X x=1 ep (axk + bxl ), p ∤ ab, 1 ≤ l < k < p − 1, ep (axk + bx−l ), p ∤ ab, 1 ≤ l ≤ k, (k + l) < p − 1. In [8] we established the Mordell type bound |S± (k, l)| ≤ p1/4 M± (k, l)1/4 , (25) and the elementary bounds ([8, Lemma 3.2]) (26) M+ (k, l) ≤ kl(p − 1)2 , 2 M− (k, l) ≤ 3kl(p − 1) , for 1 ≤ l < k < p − 1, for 1 ≤ l ≤ k, l + k < p − 1, from which we immediately deduce Lemma 5.1. For any k, l, |S+ (k, l)| ≤ (kl)1/4 p3/4 , |S− (k, l)| ≤ (3kl)1/4 p3/4 . Set d = (k, l, p − 1), l+ = l, d1 = (k, l), l− = 2l, δ+ = In [7, Lemma 3] we proved that if k < d∗ = d∗± = (k ∓ l, p − 1) (k − l) , d1 1 32 (p 2 δ− = 1 (k + l) . d1 1 − 1) 3 d16 l±6 , then M± (k, l) ≤ d2 (p − 1)2 + 2k 2 l± (p − 1) + (p − 1)2 µ where −1 µ = max{768 · 52/3 kl± δ± 3 d/d1 , 557δ± d}. In the next section we prove a version with substantially improved constants. Theorem 5.1. If 2 (k + l)5 δ± < 2.1 (kl± /d1 )(p − 1)4 (27) then M± (k, l) ≤ d2 (p − 1)2 + 2k 2 l± (p − 1) + (p − 1)2 µ with µ=  27
7 1/6 (kl± /d1 ) d, 1/3 50 δ±
1/2  7 81 4 δ± d, 503/10 if (kl± /d1 ) ≥ 3 2 if (kl± /d1 ) ≤ 3 2
7 1/3 4/3 δ± , 50
1/3 4/3 7 δ± . 50 12 TODD COCHRANE AND CHRISTOPHER PINNER 1 1 2 1 Note that condition (27) certainly holds if k ≤ 12 (1.05) 6 (p − 1) 3 d16 l±6 . From Theorem 5.1 we readily obtain an effective form of Theorem 1.1 and Lemma 1.1 in [7]. Corollary 5.1. If 1 (p − 1)2/3 d1/3 , 2 o n −1/3 k(p − 1)2 M± (k, l) ≤ 19.74 max 1, l± ∆± (28) k< then and n o1/4 −1/3 k 1/4 p3/4 , S± (k, l) ≤ 2.11 max 1, l± ∆± where ∆± = (k ∓ l)/d. Proof. The bound for S± (k, l) follows at once from the bound on M± (k, l) by (25), so it suffices to prove the latter. We may assume that (k ∓ l) > (19.74/1.5)3 d, else the bound is trivial by (26). By (28) we certainly have (p − 1)2/3 d1/3 > 2k > (k ∓ l) > (19.74/1.5)3 d, so (p − 1) > (19.74/1.5)9/2 d. Hence 2  d5/3 (k ∓ l)1/3 d2 (p − 1)2 d d5/3 1.5 = ≤ ≤ 0.006, ≤ −1/3 2/3 l 2 kl 19.74 l (k ∓ l) ± kl± ∆± (p − 1) and 2k 2 l± (p − 1) −1/3 kl± ∆± (p − 1)2 = 2k(k ∓ l)1/3 24/3 k 4/3 ≤ d1/3 (p − 1) d1/3 (p − 1) ≤ 24/3 ( 21 (p − 1)2/3 d1/3 )4/3 d1/9 = ≤ 1/3 d (p − 1) (p − 1)1/9  1.5 19.74 1/2 < 0.276.
7 1/3 4/3 δ± then If (kl± /d1 ) ≤ 32 50  4/3  4/3  1/3  1/3 3 7 3 7 l d1 l k ∓ l d1 1/3 k ≤ ≤ 2 k 2 50 k 2 l± d1 2 50 k2 d1  1/3  1/3 3 3 7 7 0.009 1 1 ≤ 2/3 ≤ 2/3 < ,
1/3 2/3 2/3 50 50 d 2 d k 2 d1/3 1 (19.74/1.5)3 d 2 and (k ± l) d l δ± d(p − 1)2 ≤ 1 + < 1.009. = k(p − 1)2 k d1 k Hence from Theorem 5.1 n o −1/3 −1/3 M± (k, l) ≤ (0.006 + 0.276)kl± ∆± (p − 1)2 + max 19.456kl± ∆± (p − 1)2 , 16.569 · 1.009k(p − 1)2 n o −1/3 ≤ 19.74 max kl± ∆± (p − 1)2 , k(p − 1)2 .  Finally, we need the following Lemma 5.2. With λ as in Theorem 1.1, |S± (k, l)| ≤ d∗ + λ(d/d∗ )5/8 p5/4 . Moreover, if −ba is not a d∗ power, then the term d∗ may be removed. BINOMIAL SUMS 13 Proof. We use the technique of Akulinichev [1] to average over the d∗ -th roots of unity. (p − 1)S± (k, l) = = p−1 p−1 X X y=1 x=1 p−1 p−1 X X x=1 y=1 k   (p−1) (p−1) ep a(xy d∗ )k + b(xy d∗ )±l   l(p−1) . ep (axk + bx±l )y d∗ ±l If ax + bx 6= 0 then the bound of Theorem 1.2 gives       l(p−1) l(p − 1) d(p − 1) k ±l ∗ d ≤Φ =Φ ep (ax + bx )y d∗ d∗ y=1 p−1 X 5/8 ≤ λ (d/d∗ ) 5 (p − 1)5/8 p 8 . If −bā is not a d∗ -th power, then this bound hold for all nonzero x and so, |S± (k, l)| ≤ λ(d/d∗ )5/8 p5/4 . If −bā is a d∗ -th power in Z∗p then we also have the d∗ values of x with axk + bx±l = 0, each contributing p − 1 to the sum, and we obtain   ∗ d(p − 1) (29) |S± (k, l) − d∗ | ≤ p−1−d Φ < λ(d/d∗ )5/8 p5/4 . p−1 d∗  6. Proof of Theorem 5.1 We follow the proof of Corollary 3.1 of [8]. For u = (u1 , u2 ) ∈ Z∗p 2 define C± (u) = #{x ∈ Z∗p : xk − 1 = u1 y k , x±l − 1 = u2 y ±l for some y ∈ Z∗p }. From (2.1) of [8] we have (30) M± (k, l) ≤ d2 (p − 1)2 + 2k 2 l± (p − 1) + d(p − 1) N X i=1 2 C± (ui ), where u1 , ..., uN represent the N distinct non-empty sets of x being counted as u varies, ordered so that (31) C± (u1 ) ≥ C± (u2 ) ≥ · · · ≥ C± (uN ) > 0. Observe the trivial bounds (see (2.2) and §3 of [8]) (32) N X i=1 and (33) C± (ui ) ≤ (p − 1), C± (ui ) ≤ min   (p − 1) , (kl± /d1 ) . i We begin with a more precise version of Lemma 3.1 of [8]. Define  7/2 (kl± /d1 )3/2 5 2 (p − 1). (34) T = [T1 ], T1 = 3 7 3 δ± 14 TODD COCHRANE AND CHRISTOPHER PINNER Lemma 6.1. For 2 (k + l)5 δ± < 2.1 (kl± /d1 )(p − 1)4 s (35) and s ≤ T √ (p − 1)2/5 (kl± /d1 )3/5 3/5 C± (ui ) ≤ (2.1)1/10 15 s . 1/5 δ± i≤s X Proof. We follow the proof of Lemma 3.1 of [8] but with an adjusted selection of parameters  1/5  1/5  1/5 2/5 (p − 1)1/5 δ± 9 8 5 (36) C = D = [C1 ], C1 = , 14 9 6 (kl± /d1 )1/5 s1/5 (37) B = [B1 ], B1 =  14 9 2/5  2/5  2/5 1/5 (kl± /d1 )2/5 δ± s2/5 9 6 , 8 5 (p − 1)2/5 (38) A = ⌈A1 ⌉, A1 = 7 3  3/5  3/5  2/5 1/5 (p − 1)3/5 δ± s2/5 8 6 . 9 5 (kl± /d1 )3/5 9 14 We leave the fractions unsimplified to show the dependence on (46). Analogous to restrictions (3.4) to (3.9) of [8] we require our choice to satisfy (39) A, B, C ≥ 1, (40) C(k + l) ≤ (p − 1), (41) BC 2 ≤ δ± , (42) Aδ± ≤ (p − 1),   1 D C 2 + CD + D2 s ≤ ABC 2 . 3 (43) Since we have (C + r)2 equations and D−1 X   1 1 1 (C +r)2 = C 2 D +2C (D −1)D + (D −1)D(2D −1) < D C 2 + CD + D2 2 6 3 r=0 we may replace (3.9) by (43). Restriction (3.8) was not required for the construction (only simplification of the final algebra). The slightly weaker restriction (40) can replace (3.5). From (32) and (33) we have the trivial bounds s X (44) C± (ui ) ≤ (kl± /d1 )s, i=1 and, applying Cauchy-Schwartz, s s X X C± (ui )1/2 ≤ (kl± /d1 )1/2 (p − 1)1/2 s1/2 . C± (ui ) ≤ (kl± /d1 )1/2 (45) i=1 i=1 So from (32) we may certainly assume that 1/5 √ √ (p − 1)3/5 δ± 1/10 3/5 > (2.1) 15 s ⇒ A > 56s > 7.48s, 1 (kl± /d1 )3/5 BINOMIAL SUMS 15 from (44) that 1/5 √ (kl± /d1 )2/5 δ± s2/5 > (2.1)1/10 15 ⇒ B1 > 3 2/5 (p − 1)  1/2 7 > 5.61, 2 and from (45) that 2/5 (p − 1)1/5 δ± > (2.1)1/5 15 ⇒ C1 > 15. (kl± /d1 )1/5 s1/5 So A ≥ 8, B ≥ 5, C ≥ 15 and (39) holds; moreover (46) C≥ 5 9 15 C1 , B ≥ B1 , A ≤ A1 . 16 6 8 Restriction (35) ensures (40): C(k + l) ≤ C1 (k + l) = (k + l)  10 21 1/5 2/5 δ± (p − 1)1/5 ≤ (p − 1). (kl± /d1 )1/5 s1/5 Since BC 2 ≤ B1 C12 = δ± we plainly have (41). For (42) observe that  3/5  2/5  2/5 6/5 (p − 1)3/5 δ± s2/5 9 6 7 9 9 ≤ (p − 1), Aδ± ≤ A1 δ± = 8 3 14 8 5 (kl± /d1 )3/5 as long as s ≤ T1 . Since C = D restriction (43) amounts to 37 Cs ≤ AB and we have 7C 7 C1 s< s = A1 ≤ A. 3B 3 (5B1 /6) Hence as in Lemma 3.1 of [8] we can deduce that s X i=1 C± (ui ) ≤ A(kl± /d1 ) + (B − 1)(p − 1) + Ck + Cl D A(kl± /d1 ) + B(p − 1) C 9 A (kl /d 1 ± 1 ) + B1 (p − 1) ≤ 8 15 16 C1



9 3/5 9 2/5 6 2/5 7 14 2/5 (kl± /d1 )3/5 s3/5 14 8 5 3 + 9 (p − 1) =


1/5 1/5 1/5 1/5 15 9 8 5 δ± 16 14 9 6   8 (p − 1)2/5 (kl± /d1 )3/5 s3/5 3/5 = 2.1 . 1/5 3 δ± √
8 1/10 15. 3 = 4.1619... < 4.1712... = (2.1) ≤ Note that 2.13/5 Theorem 5.1 will follow at once from (30) and the following lemma: Lemma 6.2. For (47) 2 (k + l)5 δ± < 2.1 (kl± /d1 )(p − 1)4  16 TODD COCHRANE AND CHRISTOPHER PINNER we have X i≤N  27
7 1/6 (kl± /d1 ) (p − 1/3 50 δ± √  7 81
δ (p − 1), ± 4 503/10 2 C± (ui ) ≤ Proof. Setting 1), if (kl± /d1 ) ≥ 3 2 if (kl± /d1 ) ≤ 3 2
7 1/3 4/3 δ± , 50
1/3 4/3 7 δ± . 50 √ (p − 1)2/5 (kl± /d1 )3/5 B = (2.1)1/10 15 1/5 δ± Lemma 6.1 implies that for any 1 ≤ s ≤ T X (48) C± (ui ) ≤ Bs3/5 . i≤s So, putting C± (ui ) = X 3 −2/5 Bi + wi , W (s) = wi , 5 i≤s for s ≤ T we have, by (48) and Lemma 2.5, X 3 X −2/5 3 X −2/5 3 i ≤ Bs3/5 − B i = γ2/5 (s)B. (49) W (s) = C± (ui ) − B 5 5 5 i≤s i≤s i≤s Thus for any J ≥ 2 with J − 1 ≤ T we have N X i=1 C± (ui )2 ≤ ≤ = X i<J X i<J C± (ui )2 + C± (uJ ) i≥J C± (ui ) C± (ui ) (C± (ui ) − C± (uJ )) + C± (uJ )(p − 1) X3 i<J X 5 Bi−2/5 (C± (ui ) − C± (uJ )) + = M1 + E 1 + E 2 + E 3 , where M1 = X i<J wi (C± (ui ) − C± (uJ )) + C± (uJ )(p − 1)  X 9 9 2 B 2 i−4/5 = B 5(J − 1)1/5 − γ4/5 (J − 1) , 25 25 i<J E1 = 3 X wi i−2/5 , B 5 i<J E2 = X i<J E3 = C± (uJ ) p − 1 − X3 i<J 5 wi (C± (ui ) − C± (uJ )) , −2/5 Bi !  3/5 = C± (uJ ) p − 1 − B(J − 1)  3 + Bγ2/5 (J − 1) . 5 By (49) we have     X 1 3  W (J − 1) 1  ≤ 9 B 2 γ2/5 (J − 1). + E1 = B W (j) 2/5 − 5 25 (J − 1)2/5 1≤j≤J−2 j (j + 1)2/5 By (49) and (31) BINOMIAL SUMS E2 = W (J − 1) (C± (uJ−1 ) − C± (uJ )) + 17 X 1≤j≤J−2 W (j) (C± (uj ) − C± (uj+1 )) 3 Bγ2/5 (J − 1) (C± (u1 ) − C± (uJ )) . 5 Observing from (48) that C± (u1 ) ≤ B we then get ≤ 3 2 3 B γ2/5 (J − 1) − Bγ2/5 (J − 1)C± (uJ ). 5 5 Hence, with κ1 (J − 1) as defined in (7), E2 ≤ N X i=1 C± (ui )2 ≤    9 2 B 5(J − 1)1/5 + κ1 (J − 1) + C± (uJ ) p − 1 − B(J − 1)3/5 , 25 From Lemma 2.5 we have κ1 (J − 1) < −1.4 for any J ≥ 2, so for any 2 ≤ J ≤ T + 1 (50) N X i=1 C± (ui )2 ≤   9 2 B (J − 1)1/5 − 0.504B 2 + C± (uJ ) p − 1 − B(J − 1)3/5 , 5 where the 0.504 can be replaced by 0.8 when J = 2 using κ1 (1) = −20/9. We note from (33) the trivial bounds N X (51) i=1 and (52) N X i=1 C± (ui )2 ≤ (p − 1) C± (ui )2 ≤ (kl± /d1 ) N X i=1 N X i=1 C± (ui ) ≤ (p − 1)2 , C± (ui ) ≤ (kl± /d1 )(p − 1). We consider two cases.
7 1/3 4/3 Case 1: Suppose first that (kl± /d1 ) ≥ 32 50 δ± . Equivalently 5/3  1/6   7/2   1/3 δ± 1 50 2 (kl± /d1 )3/2 5 p−1 (p−1) ≤ (p−1) = T1 . = 3 B 15 7 (kl± /d1 ) 7 3 δ± In this situation we take J= If J = 1 then (kl± /d1 ) 1/3 δ± ≥ 1 15
50 1/6 7 & p−1 B 5/3 ' . (p−1) and the bound claimed is at least 59 (p−1)2 and trivial. Hence we may assume that 2 ≤ J ≤ T + 1. By (48) we have C± (uJ ) ≤ C± (uJ−1 ) ≤ B/(J − 1)2/5 and, using that x3/5 − (⌈x⌉ − 1)3/5 ≤ 35 (⌈x⌉ − 1)−2/5 ,   5/3 !3/5  2 B p − 1  C± (uJ )((p − 1) − B(J − 1)3/5 ) ≤ − (J − 1)3/5  B (J − 1)2/5 ( 0.6B 2 if J = 2, B2 3 ≤ ≤ 2 4/5 5 (J − 1) 0.345B if J ≥ 3. 18 TODD COCHRANE AND CHRISTOPHER PINNER Hence from (50) N X 9 2 B (J − 1)1/5 5 i=1  1/3 9 2 p−1 ≤ B 5 B  1/6 7 (kl± /d1 ) = 27 (p − 1). 1/3 50 δ±

5/3 7 1/6 4/3 Case 2: Suppose now that (kl± /d1 ) < 32 50 δ± (that is p−1 > T1 ). B From (52) we can assume that (kl± /d1 ) > 16.568δ± and from (47) that (kl/d1 )1/4 (p−
1/4 1/2 (k + l)5/4 δ± . So 1) > 10 21 T1 > .1728 C± (ui )2 ≤ 1/4 (kl± /d1 )3/2 5/4 (kl± /d1 ) (p − 1) ≥ .1728 (16.568) (p − 1) 3 7/4 δ± δ± 5/4 (kl± /d1 )1/4 (p − 1) 1/2 ((k + l)/d1 )5/4 δ±  1/4 10 5/4 5/4 5/4 ≥ .1728 (16.568) d1 > 4.79d1 , 21 ≥ .1728 (16.568) and T ≥ 4 and T ≥ 54 T1 . We take J = T + 1, where T gives 3/5 N X i=1 ≤ 3/5 T1 < (p − 1)/B. Hence, with C± (uT ) ≤ B/T 2/5 from (48), (50) 9 2 1/5 B B T − 0.504B 2 + 2/5 (p − 1 − BT 3/5 ) 5 T  9 B 4 B  3/5 2 = p − 1 − BT (p − 1) − 0.504B − 5 T 2/5 5 T 2/5 9 B < (p − 1) 5 T 2/5 √   B 81 7 9 (p − 1) = δ± (p − 1). < 5 ( 45 T1 )2/5 503/10 4 C± (ui )2 <  7. Decimations and a bound on M± (k, l) Of independent interest and as a byproduct of the proof of Theorem 1.3 we also prove the following bound on M± (k, l): Theorem 7.1. Let c = 0.59349. If (53) ∗ −1 d = (k ∓ l, p − 1) < c  p−1 d 16/23 , BINOMIAL SUMS −1 (54) (k, p − 1) < c and (55) (l, p − 1) < c then   p−1 d p−1 d 19 16/23 7/23 , , M± (k, l) ≤ 27.57 d26/23 (p − 1)66/23 . The theorem has a direct application to a conjecture of Goresky and Klapper [13] on the decimation of ℓ-sequences. Let E = {2, 4, 6, . . . , p − 1} be the set of (non-zero) even residues in Zp and O = {1, 3, 5, . . . , p − 2} the set of odd residues. If (k, p − 1) = 1 and p ∤ A then the mapping x → Axk is a permutation of Zp . Our interest is in determining when it is a permutation of E. The conjecture is essentially equivalent to the following. GK-conjecture: For p > 13, if the mapping x → Axk is a nontrivial permutation of Zp then there exists an x ∈ E such that Axk ∈ O. In [7] Bourgain, Paulhus and the authors established the conjecture for p > 2.26 · 1055 . Here we obtain, Corollary 7.1. The GK-conjecture holds for p > 4.92 × 1034 . Proof. By [7] Theorem 1 we know that the GK-conjecture holds as long as M = M+ (k, 1) < 0.000823p3 . If d∗ ≤ 1.62p16/23 then (d = 1 and (k, p − 1) = 1) by Theorem 7.1 we have M ≤ 27.57(p − 1)66/23 and the conjecture holds for p larger than 23/3  27.57 ≤ 4.92 × 1034 . 0.000823 √ If p > 2.1 × 107 and d∗ > 1.62p16/23 then d∗ > 10 p and the result follows from Theorem 4b of [7].  8. Proof of Theorems 1.3 and 7.1 For Theorem 1.3 we need to show that |S± (k, l)| ≤ d∗ + 27.571/4 d13/46 p89/92 and for Theorem 7.1 that (subject to restrictions (53), (54),(55)) M± (k, l) ≤ 27.57 d26/23 (p − 1)66/23 . Observing the trivial bounds |S± (k, l)| ≤ p, and M± (k, l) ≤ d(p − 1)3 we may certainly assume that p > 27.5723/3 d26/3 (56) for Theorem 1.3 and p − 1 > 27.5723/3 d (57) for Theorem 7.1. Make a change of variables x → xm with m chosen so that (58) mk ≡ α mod (p − 1), ±ml ≡ β mod (p − 1), 20 TODD COCHRANE AND CHRISTOPHER PINNER (plus sign for S+ (k, l) or M+ (k, l) and minus for S− (k, l) or M− (k, l)) with (59) 0≤α≤ 1 7/23 d (p − 1)16/23 , c |β| ≤ cd16/23 (p − 1)7/23 , c = 0.59349, (α, β) 6= (0, 0). Such a pair (α, β) exists since the set of all (α, β) satisfying (58) is a lattice of volume d(p − 1) (or one can apply Dirichlet’s box principle.) Set λ = (α, β, p − 1), λ1 = (α, β). and ( |β| if β > 0, β = 2|β| if β < 0, (α − β) = λ1 ′ ( δ+ δ− if β > 0, if β < 0. Suppose first that α, β 6= 0, α 6= β. We will establish that for the pair (α, β) we have M (α, β) ≤ 27.57 d26/23 p66/23 . (60) From Lemma 1 of [7] we know that M± (k, l) ≤ M (α, β) and Theorem 7.1 is clear. Suppose that (m, p − 1) = ν and write Z∗p /(Z∗p )m = {w1 , ..., wν } so that (61) S± (k, l) = p−1 ν X
1X ep awik xα + bwi±l xβ . Si (α, β), Si (α, β) = ν i=1 x=1 Since α, β 6= 0, α 6= β the inner sum Si (α, β) in (61) is a genuine binomial sum. Thus by (25) and (60) 26 1 66 1 1 |Si (α, β)| ≤ 27.571/4 d 23 4 p 23 4 p 4 ≤ 27.571/4 d13/46 p89/92 , and |S± (k, l)| ≤ 27.571/4 d13/46 p89/92 , proving Theorem 1.3. We consider separately the three cases: Case 1: α ≤ 10000|β|, 2 Case 2: α > 10000|β| and (α + |β|)5 δ± ≥ 2.1 (αβ ′ /λ1 )(p − 1)4 , 5 2 Case 3: α > 10000|β| and (α + |β|) δ± ≤ 2.1 (αβ ′ /λ1 )(p − 1)4 , Case 1: From (26), (59) and (57) or (56) M (α, β) ≤ 3α|β|(p − 1)2 ≤ 30000|β 2 |(p − 1)2 ≤ 30000c2 d32/23 (p − 1)60/23 = 30, 000c2 26/23 30, 000c2 26/23 66/23 d (p − 1)66/23 < 13.91d26/23 (p − 1)66/23 . d (p − 1) <
p−1 6/23 27.572 d In Cases 2 to 4 we have α > 10000|β| and 0.9999 α α ≤ δ+ ≤ , λ1 λ1 α α ≤ δ− ≤ 1.0001 . λ1 λ1 Case 2: In this case we have β′ ≤ 1 (α + |β|)5 λ1 2 1.00017 α6 , δ ≤ ± 2.1 α(p − 1)4 2.1 (p − 1)4 λ1 BINOMIAL SUMS 21 and, using that d | λ1 , 3 ′ 3 1.00017 α7 αβ (p − 1)2 ≤ · 2 2 2.1 d(p − 1)2 7 3 1.0001 26/23 ≤ · d (p − 1)66/23 < 27.561d26/23 (p − 1)66/23 . 2 2.1 c7 Case 3: Here we can apply Theorem 5.1 to obtain M (α, β) ≤ M (α, β) ≤ λ2 (p − 1)2 + 2α2 β ′ (p − 1) + (p − 1)2 µ, where ( µ ≤ max 19.456 Since β′ 1/3 δ± and λ ≤ λ1 ≤ |β|, we have 19.456 (αβ ′ /λ1 ) 1/3 δ± (αβ ′ /λ1 ) 1/3 δ± ≤ λ ≤ 19.456 · 2α2/3 |β| d λ, 16.569 δ± λ . 2|β| , (α/λ1 )1/3 λ 2/3 λ1 2/3 26/23 ≤ 19.456 · 2c ) ≤ 19.456 · 2α2/3 |β|4/3 (p − 1)20/23 < 27.4806d26/23 (p − 1)20/23 , while using (57) 16.569δ± λ < 16.569 · 1.0001α ≤ 16.569 · 1.0001c−1 d7/23 (p − 1)16/23 = 16.569 · 1.0001c−1 ≤ 16.569 · 1.0001c−1 27.57−4/3 d3/23 (p − 1)20/23 < 0.34d3/23 (p − 1)20/23 . So (p − 1)2 µ ≤ 27.4806 d26/23 (p − 1)66/23 . From the lower bound (57) λ2 (p − 1)2 ≤ |β 2 |(p − 1)2 ≤ c2 d32/23 (p − 1)60/23 = c2 d26/23 (p − 1)66/23
p−1 6/23 d c2 ≤ d26/23 (p − 1)66/23 < 0.00047d26/23 (p − 1)66/23 , 27.572 and 2α2 β ′ (p − 1) ≤ 4α2 |β|(p − 1) ≤ < 4 30/23 d (p − 1)62/23 = c c 4
p−1 4/23 d d26/23 (p − 1)66/23 4 d26/23 (p − 1)66/23 < 0.08093 d26/23 (p − 1)66/23 . c (27.57)4/3 Hence M (α, β) < (27.4806+0.00047+0.08093)d26/23 (p−1)66/23 < 27.562d26/23 (p−1)66/23 . It remains to consider α = β or α = 0 or β = 0. (p−1) If α = β then mk ≡ ±ml mod (p − 1). So (p−1) d∗ |m and d∗ |β. In particular
16/23 (p−1) 16/23 7/23 ∗ −1 p−1 . This is ruled out in ≤ |β| ≤ cd (p − 1) and d ≥ c d∗ d Theorem 7.1 by (53). d3/23 (p − 1)20/23
p−1 4/23 d 22 TODD COCHRANE AND CHRISTOPHER PINNER For Theorem 1.3 we use Lemma 5.2, with d < (27.57)−23/26 p3/26 from (56), to get  5/8 d p5/4 |S± (k, l)| ≤ d∗ + 1.52 d∗ ≤ d∗ + 1.52c5/8 (1 − p−1 )−10/23 d195/184 p75/92 < d∗ + 1.1d13/46 d143/184 p75/92  143/184 p3/26 ∗ 13/46 ≤ d + 1.1d p75/92 27.5723/26 1.1 = d∗ + d13/46 p333/368 (27.57)11/16 < d∗ + 0.12d13/46 p89/92−1/16 . (p−1) (p−1) |m and (p−1,k) ≤ |β| ≤ cd16/23 (p − 1)7/23 , If α = 0 then (p − 1)|mk. Hence (p−1,k)
16/23 . This is ruled out in Theorem 7.1 by (54). For and so (p − 1, k) ≥ c−1 p−1 d Theorem 1.3 we have by the Weil bound for exponential sums, √ |S± (k, l)| ≤ |β| p ≤ cd16/23 p37/46 = cd13/46 (d/p3/26 )19/46 p1019/1196 c d13/46 p1019/1196 < 0.18d13/46 p89/92−3/26 . ≤ 27.5719/52 (p−1) (p−1) (p−1) (p−1,l) |m and (p−1,l) |α, and so (p−1,l) ≤
7/23 ; again ruled out in Theorem ≥ c p−1 d Similarly if β = 0 then (p − 1)|ml. So α ≤ c−1 d7/23 (p−1)16/23 . Hence (p−1, l) 7.1 by (55). For Theorem 1.3 we have, from Theorem 4.1 with λ = 1.51967...,  1/2 d |S± (k, l)| ≤ p + 1.36d15/88 p21/22 (l, p − 1) 1 ≤ 1/2 d15/23 p39/46 + 1.36 d15/88 p21/22 ≤ 1.30 d15/23 p39/46 + 1.36 d15/88 p21/22 c (1 − 1/p)7/46   1.36 (d/p3/26 )17/46 13/46 89/92 + 13/1012 =d p 1.30 p1/13 p   −17/52 (27.57) 1.36 13/46 89/92 1.30 ≤d p < 1.02d13/46 p89/92 . + (27.57)23/39 (27.57)13/132 References [1] N. M. Akulinicev, Bounds for rational trigonometric sums of a special type, (Russian) Dokl. Akad. Nauk SSSR 161 (1965), 743-745. [2] J. Bourgain, Mordell’s exponential sum estimate revistited, J. Amer. Math. Soc. 18, no. 2 (2005), 477-499. [3] J. Bourgain, Multilinear exponential sums in prime fields under optimal entropy condition on the sources, Geom. funct. anal. 18 (2009), 1477-1502. [4] J. Bourgain and M. Z. Garaev, On a variant of sum-product estimates and explicit exponential sum bounds in prime fields., Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 1, 1-21. [5] J. Bourgain, A. A. Glibichuk and S. V. Konyagin, Estimates for the number of sums and products and for exponential sums in fields of prime order, J. London Math. Soc. (2) 73 (2006), no. 2, 380-398. [6] J. Bourgain and S. V. Konyagin, Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order, C. R. Math. Acad. Sci. Paris 337 (2003), no. 2, 75-80. BINOMIAL SUMS 23 [7] J. Bourgain, T. Cochrane, J. Paulhus and C. Pinner, Decimations of L-sequences and permutations of even residues mod p, SIAM J. of Discrete Math. 23 (2009), no. 2, 842-857. [8] T. Cochrane and C. Pinner, Stepanov’s method applied to binomial exponential sums, Quart. J. Math. 54 (2003), 243-255. [9] , An improved Mordell type bound for exponential sums, Proc. Amer. Math. Soc. 133 (2005), no. 2, 313-320. [10] , Bounds on fewnomial exponential sums over Z∗p , preprint (2009). , Exponential sums over subgroups of Z∗p , preprint (2009). [11] [12] A. Garcia and J.F. Voloch, Fermat curves over finite fields, J. Number Theory 30 (1988), 345-356. [13] M. Goresky, A. Klapper, Arithmetic cross-correlations of FCSR sequences, IEEE Trans. Inform. Theory, 43 (1997), 1342-1346. [14] D.R. Heath-Brown and S.V. Konyagin, New bounds for Gauss sums derived from kth powers, and for Heilbronn’s exponential sum, Q. J. Math. 51 (2000), no. 2, 221-235. [15] S.V. Konyagin, Estimates for trigonometric sums over subgroups and for Gauss sums. (Russian) IV International Conference ”Modern Problems of Number Theory and its Applications”: Current Problems, Part III (Russian) (Tula, 2001), 86–114, Mosk. Gos. Univ. im. Lomonosova, Mekh.-Mat. Fak., Moscow, 2002. [16] S. V. Konyagin and I. E. Shparlinski, Character sums with exponential functions and their applications, Cambridge Univ. Press, Cambridge, 1999. [17] S. Mattarei, On a bound of Garcia and Voloch for the number of points of a Fermat Curve over a prime field, Finite Fields and Applications 13, no. 4, (2007), 773-777. [18] O. Moreno and F.N. Castro, On the calculation and estimation of Waring number for finite fields, Séminaires et Congrès 11 (2005), 29-40. [19] A. Weil, Number of solutions of equations in finite fields, Bull. AMS 55 (1949), 497-508. [20] Hong Bing Yu, Estimates for complete exponential sums of special types, Math. Proc. Cambridge Philos. Soc. 131 (2001), no. 2, 321-326. Department of Mathematics, Kansas State University, Manhattan, KS 66506 E-mail address: cochrane@math.ksu.edu Department of Mathematics, Kansas State University, Manhattan, KS 66506 E-mail address: pinner@math.ksu.edu