EXPLICIT BOUNDS ON MONOMIAL AND BINOMIAL EXPONENTIAL SUMS

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1 EXPLICIT BOUNDS ON MONOMIAL AND BINOMIAL EXPONENTIAL SUMS TODD COCHRANE AND CHRISTOPHER PINNER Abstract. Let p be a prime an e p = e 2πi /p. First, we make explicit the monomial sum bouns of Heath-Brown an Konyagin: p 1 x=1 epax min{λ 5/8 p 5/8, λ 3/8 p 3/ }, where λ = 2/ 3 = Secon, letting = k, l, p 1, we obtain the explicit binomial sum boun p 1 x=1 epaxk + bx l k l, p /6 p 89/92, for any nonconstant binomial ax k + bx l on Z p, by sharpening the estimate for the number of solutions of the system x k 1 + xk 2 = xk 3 + xk, xl 1 + xl 2 = x l 3 +xl. Finally, we apply the latter estimate to establish the Goresky-Klapper conjecture on the ecimation of l-sequences for p > Introuction For prime p an polynomial fx over Z, let Sf enote the exponential sum, Sf = e p fx, x=1 where e p = e 2πi /p is the aitive character on Z p. The nee for precise numeric estimates for such sums has become apparent in many areas of mathematics. For instance, to quantify the istribution of k-th powers mo p one nees estimates for the monomial sum Sx k an the binomial sum Sx k +bx. Such estimates were use by Bourgain, Paulhus an the authors [7], to resolve the Goresky-Klapper conjecture on the ecimation 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 valiity of the conjecture for p > A computer has verifie the conjecture for p < In orer to close this gap it is useful to have more precise estimates for a binomial sum. In this paper we obtain numeric estimates for Sf in the cases where f is a monomial or binomial. In particular the bouns obtaine allow us to establish the Goresky-Klapper conjecture for p > First we make explicit the monomial exponential sum bouns of Heath-Brown an Konyagin [1]. Theorem 1.1. For any prime p, multiplicative subgroup A of Z p an integer a with p a, we have, with λ = 2/ 3 = , p 1/2, A > 1 3 p2/3 e p ax λ A x A 3/8 p 1/, p 1/2 < A 1 3 p2/3 λ A 5/8 p 1/8, 3 p 1/3 < A p 1/2. Date: October 5,

2 2 TODD COCHRANE AND CHRISTOPHER PINNER Equivalently, letting A Z p be the subgroup of -th powers we have Theorem 1.2. Let p be a prime an a positive integer with p 1. Then for any integer a with p a, p 1 1p 1/2 + 1, < 3 p 1/3, e p ax λ x=1 5/8 p 5/8, 3 p 1/3 < p 1/2, λ 3/8 p 3/, p 1/2 < 1 3 p2/3, with λ = 2 3 = Each of the bouns in Theorems 1.1 an 1.2 is vali for arbitrary. We have inicate to the right the interval where the boun is optimal. The first boun in each of these theorems is just the classical boun for a Gauss sum. For A < 3p 1/3, or equivalently > 1 3 p2/3 all of these bouns are trivial. Konyagin [15] has obtaine nontrivial bouns for A > p 1 +ɛ that can be mae explicit, but he an we have not compute the constants. Bourgain an Garaev [] also have the boun x A e pax A for any A with A > p 1/, but the implie constant has not been compute. Bourgain an Konyagin [6], an Bourgain, Glibichuck an Konyagin [5] obtaine estimates vali for A > p ɛ. More recently Bourgain [3] has prove that log p exp C e p ax < p log A A x A for some absolute unetermine constant C > 1. For example, to save a factor e log p on the trivial boun one nees only log A > 2C log p/ log log p. Next, we turn to binomial sums. Let a, b, k, l be integers an fx = ax k + bx l. We shall only insist that f be nonconstant on Z p. Thus, it is allowe to collapse to a nonconstant monomial. Set = k, l, p 1. In [8, Corollary 1.1] the authors establishe the upper boun e p ax k + bx l k l, p 1 + 3/13 p 51/52, x=1 for any nonconstant binomial fx on Z p, a boun that is nontrivial for p 1/12. Here we establish a stronger an explicit boun, that remains nontrivial for p 3/26. Theorem 1.3. For any nonconstant binomial fx on Z p we have e p ax k + bx l k l, p /6 p 89/92. x=1 The first term can be remove if ba is not a k l-th power in Z p. The term := k l, p 1 cannot be remove from the right-han sie when bā is a -th power. Inee, in this case we see that Sf if min{ 1/2 p 3/, 5/13 p 10/13 } by estimate 29 an the bouns in Theorem 1.2. In Theorem.1 we give a slightly stronger upper boun, nontrivial for < p 1/3 but requiring k, p 1 or l, p 1 to be sufficiently large. By the work of Bourgain [2],

3 BINOMIAL SUMS 3 it is known that if < p 1 ɛ an < p 1 ɛ, then Sf p 1 δ for some δ = δɛ; see [11]. Crucial for our binomial bouns will be estimates for Mk, l, the number of solutions in Z p of the system of simultaneous equations x k 1 + x k 2 = x k 3 + x k x l 1 + x l 2 = x l 3 + x l. For example when the exponents 1 l < k have kl, p 1 = 1 an k l, p p 1 16/23 we obtain the boun see Theorem 7.1 Mk, l 27.57p 66/23. The special case l = 1 of this will be use in Corollary 7.1 when verifying the Goresky-Klapper conjecture, replacing the in Theorem 3 of our earlier paper [7]. Goo bouns for Mk, l translate immeiately into goo bouns for the corresponing binomial exponential sum via e p ax k + bx l Mk, l1/ p 1/. x=1 We shoul remark that the various inequalities above in fact hol for the more general mixe exponential sum p 1 x=1 χxe pfx, where χ is a multiplicative character mo p an f is a monomial or binomial. 2. Proofs of Theorems 1.1 an 1.2 Define NA = {x 1, x 2, y 1, y 2 A : x 1 + x 2 = y 1 + y 2 }, an for any a F p let NA, a = {x 1, x 2 A 2 : x 1 + x 2 = a}. In orer to pass from the estimate of NA to the estimate of the monomial exponential sum, we use Lemma 2.1. [1] For any subgroup A of Z p, { e p ax NA 1 A 1 p 1, NA 1 p 1 8. x A If A p 2/3 then Theorem 1.1 follows from the classical estimate for a Gauss sum, x A e pax p. For A < p 2/3 it is an immeiate consequence of Lemma 2.1 an Theorem 2.1. For any multiplicative subgroup A of Z p with A < p 2/3 we have NA 16 3 A 5/2. The classical estimate of Hua, Vaniver an Weil for the number of solutions of the homogeneous equation x 1 + x 2 = x 3 + x can be state in the manner NA A p < p A. Thus for A p2/3 one has NA < A p + A 5/2. One also sees that the hypothesis A < p 2/3 in the theorem cannot be relaxe. 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 x + y = z over

4 TODD COCHRANE AND CHRISTOPHER PINNER a finite fiel. It is a refinement of a result of Garcia an Voloch [12]. A similar upper boun is also given in [16] with an unetermine constant. Lemma 2.2. For any nonzero a Z p an multiplicative subgroup A of Z p with A < 1/ 1/ p 1 3/, we have NA, a 3 2 2/3 A 2/3. The result of [17] has an extra hypothesis that, where = p 1/ A, but one can check that the lemma hols trivially for <. Inee, for = 3, NA, a A 3 2 2/3 A 2/3 provie that A 6. If A 7 then since p = 3 A we have A = p 1/3 1/ 1/ p 1 3/, contrary to assumption. A similar argument applies when = 1 or 2. In [12] the upper boun A 2/3 is obtaine for A < p 1/p 1 1/ + 1. Let A be a multiplicative subgroup of Z p, with t = A. We start by writing A + A as a isjoint union of cosets of A, where {0} is omitte if 1 / A. For any coset Ax j let A + A = Ax 1 Ax 2 Ax n {0}, N j = {x A : x + 1 Ax j } = {x, y A A : x + y = x j }. We assume the sets Ax i have been orere so that N 1 N 2 N 3 N n. Now for any x A, x 1, x + 1 Ax j for some j an so n 1 N j = t δ, where an δ = j=1 { 1, if 1 A, 0, if 1 / A, 2 NA = δt 2 + t n Nj 2. The next lemma is extracte from the proof of [16, Lemma 3.2]. Lemma 2.3. Let a, b,, s be positive integers such that s n, sa + 1 2s 1 < ab 2, ab t, tb < p, where t = A. Then s a 1 + 2tb 1 N j. j=1 Proof. The lower case a, b, in the lemma correspon to the upper case A, B, D in [16]. In equation 3.11 of [16] one actually has sa s 1 < ab2 by summing over k in the preceing line of their proof. Then Apply the lemma with j=1 b = [st 1/3 ] + 1, a = [t/b], = 2a. sa s 1 < a2 s = aas t b as ab2,

5 BINOMIAL SUMS 5 an so if tb p then we euce s 3 j=1 N j tb a a 2 b 1 = b2 b 1 b. t + tb 1 t If we assume further that b 2 < t we get from 3, s N j b2. j=1 If b 2 t then the same boun hols trivially by 1. Since the left-han sie is an integer the 1 2 can be roppe, thus establishing Lemma 2.. For any positive integer s n such that bt < p, s N j [ts 1/3 ] j=1 Sections 5 an 8 will require us to asymptotically evaluate sums of the form j s j c. Hence for 0 < c < 1 we efine 5 γ c s = c 1 c + c s In 5 we will nee estimates for the quantity 6 κ 0 s = 21/3 9 γ 2/3s + an in 8 1 {x}x 1 c x. 2 5/ κ 1 s = 8 3 γ 2/5s γ /5 s. Lemma 2.5. For 0 < c < 1 an s in N 8 j c = s1 c 1 c γ cs. j s γ 1/3 s, The functions κ 0 s an κ 1 s are increasing for s in N with 9 κ 0 s < 2.083, an 10 κ 1 s < 1. for all s in N, with 11 κ 1 1 = Proof. Partial summation gives 8. Claims 9 an 10 follow from κ 0 s = κ 0 25/3 {x} /3 x 1/3 16 x, 27 s x 5/3 κ 1 s = κ 1 {x} 5 x 9/5 3 x2/5 1 x, s checking numerically that κ 0 1 < κ 0 2 an numerical computation κ 0 < an κ 1 < 1..

6 6 TODD COCHRANE AND CHRISTOPHER PINNER 3. Proof of Theorem 2.1 Suppose t < p 2/3. Since NA t t5/2 for t 28 we may assume that t 29 an p 157. Hence t < p 2/3 <.7p 1 3/, an by 2 an 1 an Lemma 2.2 we have NA δt 2 + tn 1 n j=1 N j = δt 2 + tn 1 t δ t /3 t 5/3 t 1 < 16 3 t5/2 for t 85. Hence we assume that t 86 an, setting [ ] J = t/, that J 5. We efine From Lemma 2.2 giving m j = 27/3 3 t2/3 j 1/3, w j = N j m j, Cs = w j. j s Cs = j s N j j s m j t 2/ /3 s 27/3 3 j s j 1/3 12 C t 2/3, C t 2/3, C3 1.92t 2/3. For s J 1 1 t1/2 1 we have b 2 1/3 t 1/3 1 t1/2 1 1/ < t,, bt < t 3/2 < p, an Lemma 2. gives j s N j [ 2 2/3 t 1/3 s 1/3] Hence, using the notation 5 an Lemma 2.5, for s J 1 13 Cs an by Lemma 2. [ 2 2/3 t 1/3 s 1/3] 2 2 7/ /3 t 1/3 s 1/ /3 3 j 1/3 3 t2/3 j s 3 2 s2/3 γ 1/3 s t 2/3 = 27/3 3 γ 1/3st 2/ /3 t 1/3 s 1/ /3 3 γ 1/3J 1t 2/3 + 2t 1/2, [ s 2 1 N s s 1 2/3 t 1/3 s 1/3] N i. s

7 BINOMIAL SUMS 7 Using 1 an 2 we write n NAt 1 = Nj 2 + δt j=1 j<j Nj 2 + N J N j + δt j J = j<j N j N j N J + N J t δ + δt j<j m j N j N J + j<j w j N j N J + N J t 1 + t = m 2 j + w j N j N J + m j w j + N J t 1 m j + t j<j j<j j<j j<j = M 1 + E 1 + E 2 + E 3 + t, where an M 1 = m 2 j = 21/3 9 t/3 j 2/3 j<j j<j = 21/3 9 t/3 3J 1 1/3 γ 2/3 J 1 = 16 3 t3/2 21/3 9 γ 2/3J 1t /3 21/3 3 t/3 E 3 = N J t 1 m j j<j t 1/3 J 1 1/3 = N J t 1 27/3 3 3 t2/3 2 J 12/3 γ 1/3 J 1 t 2 7/3 2/3 = N J 3 γ 1/3J 1t 2/3 + 2 /3 t 2/3 J 1 2/3 Using partial summation eg Hary & Wright Theorem 21, 13 an N /3 t 2/3, E 1 = w j N j N J = CJ 1N J 1 N J + CjN j N j+1 j<j 1 j J 2 2 7/3 3 γ 1/3J 1t 2/3 + 2t 1/2 N j N j+1 1 j J 1 2 7/3 = 3 γ 1/3J 1t 2/3 + 2t 1/2 N 1 N J 2 2 5/3 γ 1/3 J 1t / /3 t 7/6 7/3 N J 3 γ 1/3J 1t 2/3 + 2t 1/2. 1.

8 8 TODD COCHRANE AND CHRISTOPHER PINNER Similarly, using the bouns 12 on Cj for j 3 an 13 for j, E 2 = m j w j = 27/3 3 t2/3 w j j 1/3 j<j j<j = 27/3 3 t2/3 CJ 1J 1 1/3 + 1 j J 2 Cj j 1/3 j + 1 1/3 27/3 3 t/ / /3 3 1/ /3 1/3 + 27/3 2 7/3 3 t2/3 3 γ 1/3J 1t 2/3 + 2t 1/2 J 1 1/3 + j 1/3 j + 1 1/3 j J γ 1/3J t /3 + 28/3 3 t7/6. Hence where, with κ 0 J 1 as efine in 6, an E 5 = N J Also, from 1, 2 /3 t NAt t3/2 + t /3 E + E 5 E = κ 0 J t 1/6 + t 1/3, 2/3 J 1 2/3 t 2/3 2t 5/6 t /3 21/3 3 [ 2 2/3 J 1 1/3 t 1/3] N J N J 1. J 1 For t < 270 one checks numerically that E + E 5 <.73, whence NAt t3/2. For t 270 we have J 13. The bouns 15 N J 22/3 J 1 1/3 t 1/ J 1 an, using 15, 2/3 t J 1 2/3 = J 1 1/3 + 22/ / /3 2 J 1 1/3 t 2/ t2/3 J 1 1/3, 1/3 t J 1 1/3 2 3 J 1 2/3, t 1/3 t 1/3 J 1 1/3 2J 1 1/ /3 t 3 J 1 2/3 J 1 1/3 t 1/ J 1 1/3 J 1 1/3, t 1/3 J 1 1/3.

9 BINOMIAL SUMS 9 give E / / J 1 2/3 5.22t 1/6 J 1 1/3 = /3 t J 1 1/ t 1/6 J 1 1/3 2/ /3 2/3 1/3 t t 5.22 t 1/3 J 1 J 1 t 1/3 J 1 J 1 2/3 1/ t 1/3 < 0.760t 1/ From Lemma 2.5 we have κ 0 J 1 < Hence for t 270 we have E + E t 1/ t 1/3 < 0, an NAt 1 < 16 3 t3/2.. Another Binomial Sum Boun The following theorem is neee in the proof of Theorem 1.3, but it has inepenent interest. It yiels a nontrivial boun on any binomial exponential sum with p 1/3 an either k, p 1 > or l, p 1 >, where = k, l, p 1. Theorem.1. For any nonconstant binomial fx = ax k + bx l, an constant λ as in Theorem 1.2, we have the boun 1/2 Sf p + min{λ 8/11 15/88 p 21/22, λ 2/3 1/8 p 23/2 }. k, p 1 The proof uses averaging methos similar to what is foun in Akulinicev [1], Yu [20] an the author s work [10], together with the bouns for a monomial sum given in Theorem 1.2. For any integer k, set 16 Φk = max a 0 e p ax k. Lemma.1. For any binomial fx = ax k + bx l, we have Sf Φ p 1 l,p 1. In particular, with λ as in Theorem 1.2, x= Sf p3/2 l, p 1, l, p 1 > 1 3 p2/3, Sf λ p5/ 5/8 l, p 1 5/8, p < l, p 1 < 1 3 p2/3, Sf λ p9/8 3/8 l, p 1 3/8, l, p 1 < p. The inequality in 18 is a generalization of Yu [20, Theorem 2]. His theorem require l p 1 an = 1. From this he euce the uniform boun Sf p 23/2 uner the same constraints.

10 10 TODD COCHRANE AND CHRISTOPHER PINNER Proof. Set m = p 1 l,p 1. Then p 1Sf = e p fxy m = e p ax k y km + bx l y=1 x=1 x=1 y=1 an so, Sf 1 p 1 e p ax k y km. x=1 y=1 The first result follows from the observation that km, p 1 = remaining inequalities are an immeiate consequence of Theorem 1.2. p 1 l,p 1. The In [10, Lemma 3.1] the authors prove We euce from Theorem 1.2, 1/2 Sf p + p Φl, p 1 1/2. k, p 1 Lemma.2. For any nonconstant binomial fx = ax k + bx l on Z p, 20 1/2 Sf p k,p 1 + p 3/ l, p 1 1/2, l, p 1 < 3p 1/3, 21 1/2 Sf p k,p 1 + λ 1/2 p 13/16 l, p 1 5/16, 3p 1/3 l, p 1 < p 1/2, 22 1/2 Sf p k,p 1 + λ 1/2 p 7/8 l, p 1 3/16, p 1/2 l, p 1 < 1 3 p2/3, with λ as in Theorem 1.2. Proof of Theorem.1. We treat a number of separate cases which may be of inepenent interest. The theorem itself just nees the argument presente in cases iv an v. i. If l, p 1 > 1 3 p2/3 then by 17, Sf < 3p 5/6. ii. If p < l, p p2/3 then by 18, Sf < λp 15/16. iii. If l, p 1 < 3p 1/3 then by 20, Sf < A + p 3/ 3p 1/3 1/2 < A + 3p 11/12, where A is the first term in the theorem. iv. Suppose next that 3p 1/3 l, p 1 p. If l, p 1 λ 8/11 p 5/11 6/11 we use 19 to get Sf λ 8/11 15/88 p 21/22. If l, p 1 λ 8/11 p 5/11 6/11 then we use 21 to get the same boun with A ae. v. Suppose that p l, p p2/3. If l, p 1 > λ 8/9 2/3 p /9 then use 19 to get Sf λ 2/3 1/8 p 23/2. If l, p 1 λ 8/9 2/3 p /9, then we use 22 to get the same with A ae.

11 BINOMIAL SUMS Lemmas for Theorem 1.3 For any integers k, l let Mk, l enote the number of solutions in Z p of the system x k 1 + x k 2 = x k 3 + x k x l 1 + x l 2 = x l 3 + x l, an put M + k, l = Mk, l for 1 l < k < p 1, M k, l = Mk, l for 1 l < k, k + l < p 1. Let 23 S + k, l = e p ax k + bx l, p ab, 1 l < k < p 1, an x=1 2 S k, l = e p ax k + bx l, p ab, 1 l k, k + l < p 1. x=1 In [8] we establishe the Morell type boun 25 S k, l p 1/ M k, l 1/, an the elementary bouns [8, Lemma 3.2] 26 M + k, l klp 1 2, for 1 l < k < p 1, M k, l 3klp 1 2, for 1 l k, l + k < p 1, from which we immeiately euce Lemma 5.1. For any k, l, Set S + k, l kl 1/ p 3/, S k, l 3kl 1/ p 3/. = k, l, p 1, 1 = k, l, = = k l, p 1 l + = l, l = 2l, δ + = k l k + l, δ =. 1 1 In [7, Lemma 3] we prove that if k < 1 32 p l 1 6, then M k, l 2 p k 2 l p 1 + p 1 2 µ where µ = max{ /3 kl δ 1 3 / 1, 557δ }. In the next section we prove a version with substantially improve constants. Theorem 5.1. If 27 k + l 5 δ 2 < 2.1 kl / 1 p 1 then with M k, l 2 p k 2 l p 1 + p 1 2 µ µ = 7 1/2 1/6 kl / 1 δ 1/ /10, if kl / δ, if kl / /3 δ /3, /3 δ /3.

12 12 TODD COCHRANE AND CHRISTOPHER PINNER Note that conition 27 certainly hols if k p l 1 6. From Theorem 5.1 we reaily obtain an effective form of Theorem 1.1 an Lemma 1.1 in [7]. Corollary 5.1. If 28 k < 1 2 p 12/3 1/3, then an where = k l/. { } M k, l 19.7 max 1, l 1/3 kp 1 2 { } 1/ S k, l 2.11 max 1, l 1/3 k 1/ p 3/, Proof. The boun for S k, l follows at once from the boun on M k, l by 25, so it suffices to prove the latter. We may assume that k l > 19.7/1.5 3, else the boun is trivial by 26. By 28 we certainly have p 1 2/3 1/3 > 2k > k l > 19.7/1.5 3, so p 1 > 19.7/1.5 9/2. Hence 2 p 1 2 kl 1/3 p 1 = 5/3 k l1/3 2 kl an 2k 2 l p 1 2kk l1/3 = kl 1/3 p 1 2 1/3 p 1 2/3 k /3 1/3 p 1 5/3 k l 2/3 l /3 1 2 p 12/3 1/3 /3 = 1/3 p 1 If kl / /3 / δ then l k 3 1/3 /3 7 1 l k l 2 50 k 2 3 1/3 7 1 k l k 2 21/ / /3 50 1/3 k 3 1/ /3 2 2/3 50 an δ p 1 2 kp 1 2 = Hence from Theorem 5.1 k l k 2 l 0.006, 1/9 1/2 1.5 p 1 < / /3 1/ /1.53 2/3 < 1 + l 1 k < M k, l kl 1/3 p max { 19.7 max kl 1/3 p 1 2, kp 1 2}. Finally, we nee the following Lemma 5.2. With λ as in Theorem 1.1, S k, l + λ/ 5/8 p 5/. Moreover, if ba is not a power, then the term may be remove , { 19.56kl 1/3 p 1 2, kp 1 2}

13 BINOMIAL SUMS 13 Proof. We use the technique of Akulinichev [1] to average over the -th roots of unity. p 1S k, l = e p y=1 x=1 x=1 y=1 axy p 1 k + bxy p 1 l = e p ax k + bx l y lp 1. If ax k + bx l 0 then the boun of Theorem 1.2 gives e p ax k + bx l y lp 1 lp 1 p 1 Φ = Φ y=1 λ / 5/8 p 1 5/8 p 5 8. If bā is not a -th power, then this boun hol for all nonzero x an so, S k, l λ/ 5/8 p 5/. If bā is a -th power in Z p then we also have the values of x with ax k + bx l = 0, each contributing p 1 to the sum, an we obtain p 1 29 S k, l p 1 p 1 Φ < λ/ 5/8 p 5/. 6. Proof of Theorem 5.1 We follow the proof of Corollary 3.1 of [8]. For u = u 1, u 2 Z p 2 efine C u = #{x Z p From 2.1 of [8] we have : x k 1 = u 1 y k, x l 1 = u 2 y l for some y Z p}. 30 M k, l 2 p k 2 l p 1 + p 1 N Cu 2 i, where u 1,..., u N represent the N istinct non-empty sets of x being counte as u varies, orere so that 31 C u 1 C u 2 C u N > 0. Observe the trivial bouns see 2.2 an 3 of [8] 32 N C u i p 1, an { } p 1 33 C u i min, kl / 1. i We begin with a more precise version of Lemma 3.1 of [8]. Define 3 T = [T 1 ], T 1 = 5 7/2 2 kl / 1 3/2 p δ 3

14 1 TODD COCHRANE AND CHRISTOPHER PINNER Lemma 6.1. For 35 k + l 5 δ 2 < 2.1 kl / 1 p 1 s an s T C u i 2.1 1/10 15 p 12/5 kl / 1 3/5 s 3/5. i s Proof. We follow the proof of Lemma 3.1 of [8] but with an ajuste selection of parameters 1/5 1/5 1/ p 1 1/5 δ 2/5 36 C = D = [C 1 ], C 1 = kl / 1 1/5 s, 1/5 37 B = [B 1 ], B 1 = 38 A = A 1, A 1 = / /5 8 9 δ 1/5 2/5 2/5 6 kl / 1 2/5 δ 1/5 s 2/5, 5 p 1 2/5 3/5 2/5 6 p 1 3/5 δ 1/5 s 2/5. 5 kl / 1 3/5 We leave the fractions unsimplifie to show the epenence on 6. Analogous to restrictions 3. to 3.9 of [8] we require our choice to satisfy 39 A, B, C 1, 0 Ck + l p 1, 1 BC 2 δ, 2 Aδ p 1, 3 D C 2 + CD + 13 D2 s ABC 2. Since we have C + r 2 equations an D 1 C +r 2 = C 2 D+2C 1 2 D 1D+ 1 6 D 1D2D 1 < D C 2 + CD D2 r=0 we may replace 3.9 by 3. Restriction 3.8 was not require for the construction only simplification of the final algebra. The slightly weaker restriction 0 can replace 3.5. From 32 an 33 we have the trivial bouns s C u i kl / 1 s, an, applying Cauchy-Schwartz, s 5 C u i kl / 1 1/2 s C u i 1/2 kl / 1 1/2 p 1 1/2 s 1/2. So from 32 we may certainly assume that p 1 3/5 δ 1/5 kl / 1 3/5 > 2.1 1/10 15 s 3/5 A 1 > 56s > 7.8s,

15 BINOMIAL SUMS 15 from that kl / 1 2/5 δ 1/5 s 2/5 > 2.1 1/10 1/ B p 1 2/5 1 > 3 > 5.61, 2 an from 5 that p 1 1/5 δ 2/5 kl / 1 1/5 s 1/5 > 2.11/5 15 C 1 > 15. So A 8, B 5, C 15 an 39 hols; moreover 6 C C 1, B 5 6 B 1, A 9 8 A 1. Restriction 35 ensures 0: Ck + l C 1 k + l = k + l Since BC 2 B 1 C 2 1 = δ we plainly have 1. For 2 observe that Aδ 9 8 A 1δ = / /5 10 δ 2/5 p 1 1/5 p kl / 1 1/5 s1/5 2/5 2/5 6 p 1 3/5 δ 6/5 s 2/5 p 1, 5 kl / 1 3/5 as long as s T 1. Since C = D restriction 3 amounts to 7 3Cs AB an we have 7 C 3 B s < 7 C 1 3 5B 1 /6 s = A 1 A. Hence as in Lemma 3.1 of [8] we can euce that s C u i Akl / 1 + B 1p 1 + Ck + Cl D Akl / 1 + Bp 1 C 9 8 A 1kl / 1 + B 1 p 1 15 = 16 C 1 9 3/5 9 2/5 6 2/ /5 8 1/ p 1 2/5 kl / 1 3/5 s 3/5 = 2.1 3/ /5 p 1 2/5 kl / 1 3/5 s 3/5 Note that 2.1 3/5 8 3 = < = 2.1 1/ δ 1/5. δ 1/5 Theorem 5.1 will follow at once from 30 an the following lemma: Lemma 6.2. For 7 k + l 5 δ 2 < 2.1 kl / 1 p 1

16 16 TODD COCHRANE AND CHRISTOPHER PINNER we have 27 7 C u i i N Proof. Setting 50 3/ /6 kl / 1 δ 1/3 p 1, if kl / δ p 1, if kl / B = 2.1 1/10 15 p 12/5 kl / 1 3/5 δ 1/5 Lemma 6.1 implies that for any 1 s T 8 C u i Bs 3/5. So, putting i s C u i = 3 5 Bi 2/5 + w i, W s = i s w i, for s T we have, by 8 an Lemma 2.5, 9 W s = i s C u i 3 5 B i s Thus for any J 2 with J 1 T we have N C u i 2 i<j C u i 2 + C u J i J i 2/5 Bs 3/5 3 5 B i s C u i i<j C u i C u i C u J + C u J p /3 δ /3, /3 δ /3. i 2/5 = 3 5 γ 2/5sB. where = 3 5 Bi 2/5 C u i C u J + w i C u i C u J + C u J p 1 i<j i<j = M 1 + E 1 + E 2 + E 3, M 1 = i<j 9 25 B2 i /5 = 9 25 B2 5J 1 1/5 γ /5 J 1, E 1 = 3 5 B i<j w i i 2/5, E 3 = C u J p 1 i<j By 9 we have E 1 = 3 5 B W J 1 J 1 + 2/5 By 9 an 31 E 2 = i<j w i C u i C u J, 3 5 Bi 2/5 = C u J p 1 BJ 1 3/ Bγ 2/5J 1. 1 j J 2 1 W j j 1 9 2/5 j + 1 2/5 25 B2 γ 2/5 J 1.

17 BINOMIAL SUMS 17 E 2 = W J 1 C u J 1 C u J Bγ 2/5J 1 C u 1 C u J. 1 j J 2 Observing from 8 that C u 1 B we then get E B2 γ 2/5 J Bγ 2/5J 1C u J. Hence, with κ 1 J 1 as efine in 7, N W j C u j C u j+1 C u i B2 5J 1 1/5 + κ 1 J 1 + C u J p 1 BJ 1 3/5, From Lemma 2.5 we have κ 1 J 1 < 1. for any J 2, so for any 2 J T N C u i B2 J 1 1/5 0.50B 2 + C u J p 1 BJ 1 3/5, where the 0.50 can be replace by 0.8 when J = 2 using κ 1 1 = 20/9. We note from 33 the trivial bouns 51 an 52 N N C u i 2 p 1 C u i p 1 2, N N C u i 2 kl / 1 C u i kl / 1 p 1. We consier two cases. Case 1: Suppose first that kl / /3 / δ p 1 B 5/3 = 1 15 In this situation we take 1/6 50 δ 1/3 7 kl / 1 p p 5/3 1 J =. B. Equivalently 7/2 kl / 1 3/2 p 1 = T If J = 1 then kl/ /6 δ 1/ p 1 an the boun claime is at least 9 5 p 12 an trivial. Hence we may assume that 2 J T + 1. By 8 we have C u J C u J 1 B/J 1 2/5 an, using that x 3/5 x 1 3/5 3 5 x 1 2/5, p C u J p 1 BJ 1 3/5 B 2 5/3 3/5 1 J 1 3/5 J 1 2/5 B { 3 B 2 5 J 1 0.6B 2 if J = 2, /5 0.35B 2 if J 3. δ 3

18 18 TODD COCHRANE AND CHRISTOPHER PINNER Hence from 50 N C u i B2 J 1 1/5 1/3 9 p 1 5 B2 B 1/6 7 kl / 1 = 27 p Case 2: Suppose now that kl / 1 < 3 7 1/6 / δ that is p 1 5/3 B > T1. From 52 we can assume that kl / 1 > δ an from 7 that kl/ 1 1/ p 1 > 10 1/ 21 k + l 5/ δ 1/2. So T 1 >.1728 kl / 1 3/2 δ 3 δ 1/3 p / kl / 1 1/ p / kl / 1 1/ p 1 k + l/ 1 5/ δ 1/2 1/ / 10 5/ 1 >.79 5/ 1, 21 an T an T 5 T 1. We take J = T + 1, where T 3/5 T 3/5 1 < p 1/B. Hence, with C u T B/T 2/5 from 8, 50 gives δ 7/ N C u i 2 < 9 5 B2 T 1/5 0.50B 2 + B T p 1 BT 3/5 2/5 = 9 B 5 T p 1 2/5 0.50B2 B p 1 BT 3/5 5 T 2/5 < 9 B p 1 5 T 2/5 < 9 B T p 1 = δ 2/ /10 p Decimations an a boun on M k, l Of inepenent interest an as a byprouct of the proof of Theorem 1.3 we also prove the following boun on M k, l: Theorem 7.1. Let c = If 16/23 p 1 53 = k l, p 1 < c 1,

19 BINOMIAL SUMS 19 16/23 p 1 5 k, p 1 < c 1, an 7/23 p 1 55 l, p 1 < c, then M k, l /23 p 1 66/23. The theorem has a irect application to a conjecture of Goresky an Klapper [13] on the ecimation of l-sequences. Let E = {2,, 6,..., p 1} be the set of non-zero even resiues in Z p an O = {1, 3, 5,..., p 2} the set of o resiues. If k, p 1 = 1 an p A then the mapping x Ax k is a permutation of Z p. Our interest is in etermining when it is a permutation of E. The conjecture is essentially equivalent to the following. GK-conjecture: For p > 13, if the mapping x Ax k is a nontrivial permutation of Z p then there exists an x E such that Ax k O. In [7] Bourgain, Paulhus an the authors establishe the conjecture for p > Here we obtain, Corollary 7.1. The GK-conjecture hols for p > Proof. By [7] Theorem 1 we know that the GK-conjecture hols as long as M = M + k, 1 < p 3. If 1.62p 16/23 then = 1 an k, p 1 = 1 by Theorem 7.1 we have M 27.57p 1 66/23 an the conjecture hols for p larger than 23/ If p > an > 1.62p 16/23 then > 10 p an the result follows from Theorem b of [7]. 8. Proof of Theorems 1.3 an 7.1 For Theorem 1.3 we nee to show that S k, l / 13/6 p 89/92 an for Theorem 7.1 that subject to restrictions 53, 5,55 M k, l /23 p 1 66/23. Observing the trivial bouns S k, l p, an M k, l p 1 3 we may certainly assume that 56 p > /3 26/3 for Theorem 1.3 an 57 p 1 > /3 for Theorem 7.1. Make a change of variables x x m with m chosen so that 58 mk α mo p 1, ml β mo p 1,

20 20 TODD COCHRANE AND CHRISTOPHER PINNER plus sign for S + k, l or M + k, l an minus for S k, l or M k, l with 59 0 α 1 c 7/23 p 1 16/23, β c 16/23 p 1 7/23, c = , α, β 0, 0. Such a pair α, β exists since the set of all α, β satisfying 58 is a lattice of volume p 1 or one can apply Dirichlet s box principle. Set λ = α, β, p 1, λ 1 = α, β. an β = { β if β > 0, 2 β if β < 0, { α β δ + if β > 0, = λ 1 δ if β < 0. Suppose first that α, β 0, α β. We will establish that for the pair α, β we have 60 Mα, β /23 p 66/23. From Lemma 1 of [7] we know that M k, l Mα, β an Theorem 7.1 is clear. Suppose that m, p 1 = ν an write Z p/z p m = {w 1,..., w ν } so that 61 S k, l = 1 ν ν S i α, β, S i α, β = e p aw k i x α + bw l i x β. Since α, β 0, α β the inner sum S i α, β in 61 is a genuine binomial sum. Thus by 25 an 60 S i α, β / p p x= / 13/6 p 89/92, an S k, l / 13/6 p 89/92, proving Theorem 1.3. We consier separately the three cases: Case 1: α β, Case 2: α > β an α + β 5 δ αβ /λ 1 p 1, Case 3: α > β an α + β 5 δ αβ /λ 1 p 1, Case 1: From 26, 59 an 57 or 56 Mα, β 3α β p β 2 p c 2 32/23 p 1 60/23 = 30, 000c2 6/23 26/23 p 1 66/23 < p 1 In Cases 2 to we have α > β an Case 2: In this case we have α λ 1 δ + α λ 1, 30, 000c /23 p 1 66/23 < /23 p 1 66/23. α λ 1 δ α λ 1. β 1 α + β 5 λ αp 1 δ α p 1, λ 1

21 BINOMIAL SUMS 21 an, using that λ 1, Mα, β 3 2 αβ p α 7 p c 7 26/23 p 1 66/23 < /23 p 1 66/23. Case 3: Here we can apply Theorem 5.1 to obtain where Since an λ λ 1 β, we have αβ /λ 1 δ 1/3 while using 57 Mα, β λ 2 p α 2 β p 1 + p 1 2 µ, { } µ max αβ /λ 1 λ, δ δ 1/3 λ. β δ 1/3 λ α 2/3 β 2 β α/λ 1 1/3, λ λ 2/ α 2/3 β / c 2/3 26/23 p 1 20/23 < /23 p 1 20/23, δ λ < α c 1 7/23 p 1 16/23 = c 1 3/23 p 1 20/ c /3 3/23 p 1 20/23 < 0.3 3/23 p 1 20/23. So p 1 2 µ /23 p 1 66/23. From the lower boun 57 an λ 2 p 1 2 β 2 p 1 2 c 2 32/23 p 1 60/23 = c 2 p 1 c /23 p 1 66/23 < /23 p 1 66/23, 2α 2 β p 1 α 2 β p 1 c 30/23 p 1 62/23 = Hence < c p 1 6/23 26/23 p 1 66/23 /23 26/23 p 1 66/23 c /3 26/23 p 1 66/23 < /23 p 1 66/23. Mα, β < /23 p 1 66/23 < /23 p 1 66/23. It remains to consier α = β or α = 0 or β = 0. If α = β then mk ml mo p 1. So p 1 m an p 1 β. In particular p 1 β c 16/23 p 1 7/23 an c 1 p 1 16/23. This is rule out in Theorem 7.1 by 53. p 1 /23

22 22 TODD COCHRANE AND CHRISTOPHER PINNER For Theorem 1.3 we use Lemma 5.2, with < /26 p 3/26 from 56, to get 5/8 S k, l p 5/ c 5/8 1 p 1 10/23 195/18 p 75/92 < /6 13/18 p 75/ /6 p 3/26 = / /16 13/6 p 333/368 < /6 p 89/92 1/16. 13/18 p 75/92 If α = 0 then p 1 mk. Hence p 1 p 1 p 1,k m an p 1,k β c16/23 p 1 7/23, an so p 1, k c 1 p 1 16/23. This is rule out in Theorem 7.1 by 5. For Theorem 1.3 we have by the Weil boun for exponential sums, S k, l β p c 16/23 p 37/6 = c 13/6 /p 3/26 19/6 p 1019/1196 c /52 13/6 p 1019/1196 < /6 p 89/92 3/26. Similarly if β = 0 then p 1 ml. So p 1 p 1 p 1 p 1,l m an p 1,l α, an so p 1,l α c 1 7/23 p 1 16/23. Hence p 1, l c p 1 7/23; again rule out in Theorem 7.1 by 55. For Theorem 1.3 we have, from Theorem.1 with λ = , 1/2 S k, l p /88 p 21/22 l, p 1 1 c 1/2 1 1/p 7/6 15/23 p 39/ /88 p 21/ /23 p 39/ /88 p 21/22 = 13/6 p 89/ /p3/26 17/ p 1/13 p 13/ /6 p 89/ / / /132 References < /6 p 89/92. [1] N. M. Akulinicev, Bouns for rational trigonometric sums of a special type, Russian Dokl. Aka. Nauk SSSR , [2] J. Bourgain, Morell s exponential sum estimate revistite, J. Amer. Math. Soc. 18, no , [3] J. Bourgain, Multilinear exponential sums in prime fiels uner optimal entropy conition on the sources, Geom. funct. anal , [] J. Bourgain an M. Z. Garaev, On a variant of sum-prouct estimates an explicit exponential sum bouns in prime fiels., Math. Proc. Cambrige Philos. Soc , no. 1, [5] J. Bourgain, A. A. Glibichuk an S. V. Konyagin, Estimates for the number of sums an proucts an for exponential sums in fiels of prime orer, J. Lonon Math. Soc , no. 2, [6] J. Bourgain an S. V. Konyagin, Estimates for the number of sums an proucts an for exponential sums over subgroups in fiels of prime orer, C. R. Math. Aca. Sci. Paris , no. 2,

23 BINOMIAL SUMS 23 [7] J. Bourgain, T. Cochrane, J. Paulhus an C. Pinner, Decimations of L-sequences an permutations of even resiues mo p, SIAM J. of Discrete Math , no. 2, [8] T. Cochrane an C. Pinner, Stepanov s metho applie to binomial exponential sums, Quart. J. Math , [9], An improve Morell type boun for exponential sums, Proc. Amer. Math. Soc , no. 2, [10], Bouns on fewnomial exponential sums over Z p, preprint [11], Exponential sums over subgroups of Z p, preprint [12] A. Garcia an J.F. Voloch, Fermat curves over finite fiels, J. Number Theory , [13] M. Goresky, A. Klapper, Arithmetic cross-correlations of FCSR sequences, IEEE Trans. Inform. Theory, , [1] D.R. Heath-Brown an S.V. Konyagin, New bouns for Gauss sums erive from kth powers, an for Heilbronn s exponential sum, Q. J. Math , no. 2, [15] S.V. Konyagin, Estimates for trigonometric sums over subgroups an for Gauss sums. Russian IV International Conference Moern Problems of Number Theory an its Applications : Current Problems, Part III Russian Tula, 2001, 86 11, Mosk. Gos. Univ. im. Lomonosova, Mekh.-Mat. Fak., Moscow, [16] S. V. Konyagin an I. E. Shparlinski, Character sums with exponential functions an their applications, Cambrige Univ. Press, Cambrige, [17] S. Mattarei, On a boun of Garcia an Voloch for the number of points of a Fermat Curve over a prime fiel, Finite Fiels an Applications 13, no., 2007, [18] O. Moreno an F.N. Castro, On the calculation an estimation of Waring number for finite fiels, Séminaires et Congrès , [19] A. Weil, Number of solutions of equations in finite fiels, Bull. AMS , [20] Hong Bing Yu, Estimates for complete exponential sums of special types, Math. Proc. Cambrige Philos. Soc , no. 2, Department of Mathematics, Kansas State University, Manhattan, KS aress: cochrane@math.ksu.eu Department of Mathematics, Kansas State University, Manhattan, KS aress: pinner@math.ksu.eu