DOUBLE CHARACTER SUMS OVER SUBGROUPS AND INTERVALS 4 1 MEI-CHU CHANG AND IGOR E. SHPARLINSKI 0 2 Abstract. We estimate double sums b e S (a, , )= χ(x+aλ), 1 a

0 ≥ . is arbitrary. We obtain a nontrivial estimate in a wider range of h parameters H and T. We also estimate double sums t a m Tχ(a, )= χ(a+λ+µ), 1 a

0 we have H p1/4+ε, by using the Burgess bound, see [20, ≥ Theorem 12.6], or T p1/2+ε, by using (2). ≥ 1.2. Main results. Here we obtain a nontrivial estimate in a wider range of parameters H and T. DOUBLE CHARACTER SUMS OVER SUBGROUPS AND INTERVALS 3 Theorem 1. For every fixed real ε > 0 there are some δ > 0 and η > 0 such that if H > pε and T > p1/2−δ then for the interval = 1,...,H and the multiplicative subgroup F∗ of order T, we hIave{ } G ⊆ p S (a, , ) = O(HTp−η) χ I G uniformly over a F∗ and nonprincipal multiplicative characters χ of F . ∈ p p We also obtain a similar estimate at the other end of region of H and T, namely for a very small T and H that is still below the reach of the Burgess bound (see [20, Theorem 12.6]). In fact in this case we are able to estimate a more general sums S (f, , ) = χ(x+f(λ)), χ I G x∈I λ∈G XX with a non-constant polynomial f F [X]. p ∈ Theorem 2. For every fixed real ε > 0 and integer d 1 there are ≥ some δ > 0 and η > 0 such that if T > pε and H > p1/4−δ then for the interval = 1,...,H , the multiplicative subgroup F∗ of order I { } G ⊆ p T, we have S (f, , ) = O(HTp−η) χ I G uniformly over polynomials f F [X] of degree d and nonprincipal p multiplicative characters χ of F∈. p We also give an explicit version of Theorem 1 in the case when H = p1/4+o(1) and T = p1/2+o(1), that is, when other methods just start to fail. Theorem 3. Let H = p1/4+o(1) and T = p1/2+o(1). Then for the interval = 1,...,H and the multiplicative subgroup F∗ of order T, we I { } G ⊆ p have S (a, , ) HTp−5/48+o(1) χ | I G | ≤ uniformly over a F∗ and nonprincipal multiplicative characters χ of F . ∈ p p Furthermore, we also consider double sums T (a, ) = χ(a+λ+µ), 1 a < p 1, χ G ≤ − λ,µ∈G X where both variables run over a multiplicative subgroup F∗. G ⊆ p Using recent estimates of Shkredov [28] on the so-called additive energy of multiplicative subgroups we also estimate them below the obvious range T p1/2, where T = # , given by the estimate ≥ G T (a, ) Tp1/2, χ | G | ≤ 4 M.-C. CHANGANDIGOR E. SHPARLINSKI which follows from (2). Theorem 4. Let T p2/3. Then for the multiplicative subgroup F∗ of order T, we ha≤ve G ⊆ p T19/26p1/2+o(1), if T p1/2, ≤ T (a, ) T9/13p27/52+o(1), if p1/2 < T p29/48, | χ G | ≤  ≤ Tp1/3+o(1), if p29/48 < T p2/3,  ≤ uniformly over a F∗ and nonprincipal multiplicative characters χ of F . ∈ p p Note that Theorem 4 nontrivial provided that T p13/33+ε for some ≥ fixed ε > 0. We also give an application of Theorem 4 to primitive roots modulo p with few non-zero binary digits. More precisely, let u denote the p smallest u such that there exists a primitive root modulo p with u p non-zero binary digits. It is shows in [16, Theorem 5] that u 2 p ≤ for all but o(Q/logQ) primes p Q, as Q (note that in [16] ≤ → ∞ the result is formulated only for quadratic non-residues but it is easy to see that the argument also holds for primitive roots). Instead of o(Q/logQ), can obtain a slightly more explicit but still rather weak bound on the size of the exceptional set. Here we show that Theorem 4 implies a rather strong bound on the set of primes p Q for which ≤ u 3 does not hold. p ≤ Theorem 5. For all but at most Q26/33+o(1) primes p Q, we have ≤ u 3 as Q . p ≤ → ∞ We also note that one may attempt to treat the sums S (a, , ) χ I G and T (a, ) within the general theory of double sums of multiplicative χ G characters, see [6, 7, 11, 12, 15, 21, 22, 23] and references therein. However it seems that none of the presently known results implies a nontrivial estimate in the range of Theorems 1 and 4. 2. Preparations 2.1. Notation and general conventions. Throughout the paper, p always denotes a sufficiently large prime number and χ denotes an non-principal multiplicative character modulo p. We assume that F is p represented by the set 0,...,p 1 . Furthermore, alwa{ys denot−es a} multiplicative subgroup of F∗ of G p order # = T and always denotes the set = 1,...,H . We alGso assume tIhat f F [X] is a of degIree {d 1. In}particular, p ∈ ≥ f is not a constant. DOUBLE CHARACTER SUMS OVER SUBGROUPS AND INTERVALS 5 The notations U = O(V) and U V are both equivalent to the ≪ inequality U cV with some constant c > 0 that may depend on the | | ≤ real parameter ε > 0 and the integer parameters d 1 and ν 1 and ≥ ≥ is absolute otherwise. In particular, all our estimates are uniform with respect to the polynomial f and the character χ. 2.2. Bounds of some exponential and character sums. First we recall the classical result of Davenport and Erd˝os [13], which follows from the Weil bound of multiplicative character sums, see [20, Theo- rem 11.23]. Lemma 6. For a fixed integer ν 1 and an integer R < p, we have ≥ 2ν R χ(v +r) R2νp1/2 +Rνp. ≪ (cid:12) (cid:12) vX∈Fp(cid:12)Xr=1 (cid:12) (cid:12) (cid:12) The following resu(cid:12)lt is a versio(cid:12)n of Lemma 6 with ν = 1 which is (cid:12) (cid:12) slightly more precise in this case. Lemma 7. For any set F and complex numbers α of such that p v V ⊆ α 1 for v , we have v | | ≤ ∈ V 2 χ(u+v) # p. (cid:12) (cid:12) ≪ V uX∈Fp(cid:12)Xv∈V (cid:12) (cid:12) (cid:12) Proof. Denotingbyχthec(cid:12)onjugatecha(cid:12)racterandrecallingthatχ(w) = (cid:12) (cid:12) χ(w−1) for w F∗, we obtain ∈ p 2 χ(u+v) = α α χ(u+v)χ(u+w). v w (cid:12) (cid:12) uX∈Fp(cid:12)Xv∈V (cid:12) vX,w∈V uX∈Fp (cid:12) (cid:12) If v = w the(cid:12)inner sum is(cid:12)equal to p 1. So the total contribution from (cid:12) (cid:12) − such terms is O(Mp). Otherwise, we derive χ(u+v)χ(u+w) = χ(u+v w)χ(u) − uX∈Fp uX∈Fp = χ(u+v w)χ(u) = χ 1+(v w)u−1 − − u∈F∗ u∈F∗ Xp Xp (cid:0) (cid:1) = χ(1+u) = χ(1+u) χ(1) = χ(1). − − uX∈F∗p uX∈Fp So the total contribution from such terms is O(M2) = O(Mp) and the result follows. ⊔⊓ 6 M.-C. CHANGANDIGOR E. SHPARLINSKI We also need the following bound of Bourgain [2, Theorem 1]. Lemma 8. For every fixed real ε > 0 and integer r 1 there is some ≥ ξ > 0 such that for any integers k ,...,k 1 with 1 r ≥ gcd(k ,p 1) < p1−ε, and gcd(k k ,p 1) < p1−ε, i i j − − − for i,j = 1,...,r, i = j, uniformly over the coefficients a ,...,a F , 1 r p 6 ∈ not all equal to zero, we have p−1 2πi exp a xk1 +...+a xkr p1−ξ. 1 r p ≪ x=1 (cid:18) (cid:19) X (cid:0) (cid:1) Clearly for any F F [X] and a multiplicative subgroup F∗ of ∈ p G ⊆ p order # = T we have G p−1 1 2πi 1 2πi exp F(λ) = exp F(x(p−1)/T) p−ξ. # p p 1 p ≪ G λ∈G (cid:18) (cid:19) − x=1 (cid:18) (cid:19) X X so we derive from Lemma 8: Corollary 9. For every fixed real ε > 0 and integer d 1 there is some ξ > 0 such that for T pε, uniformly over a F∗, w≥e have ≥ ∈ p 2πi exp af(λ) Tp−ξ. p ≪ λ∈G (cid:18) (cid:19) X 2.3. Bound on the number of solutions to some congruences. FirstwenotethatcombiningCorollary9withtheErd˝os-Turán inequality (see, for example, [14, Theorem 1.21]) that relates the uniformity of distribution to exponential sums, we immediately obtain: Lemma 10. For every fixed real ε > 0 and integer r 1 there is some ≥ κ > 0 such that for T pε, we have ≥ HT # λ : f(λ) b+x (mod p), where x = +O T1−κ , { ∈ G ≡ ∈ I} p uniformly over b F . (cid:0) (cid:1) p ∈ Let N( , ) be the number of solutions to the congruence I G λx y (mod p), x,y , λ . ≡ ∈ I ∈ G Some of our results rely on an upper bound on N( , ) which is I G given in [9, Theorem 1], see also [10] for some other bounds. Lemma 11. Let ν 1 be a fixed integer. Then ≥ N( , ) Ht(2ν+1)/2ν(ν+1)p−1/2(ν+1)+o(1) +H2t1/νp−1/ν+o(1), I G ≤ DOUBLE CHARACTER SUMS OVER SUBGROUPS AND INTERVALS 7 as p , where → ∞ t = max T,p1/2 . { } We also use the following bound which is due to Ayyad, Cochrane and Zheng [1, Theorem 1]. Lemma 12. Let = b +1,...,b +h for some integers p > h +b > i i i i i i J { } b 1, i = 1,2,3,4. Then i ≥ # (x ,x ,x ,x ) : x x x x (mod p) 1 2 3 4 1 2 3 4 1 2 3 4 { ∈ J ×J ×J ×J ≡ } 1 = h h h h +O (h h h h )1/2(logp)2 . 1 2 3 4 1 2 3 4 p (cid:16) (cid:17) We now fix some real L > 1 and denote by the set of primes of L the interval [L,2L]. We need an upper bound on the quantity W = # (u ,u ,ℓ ,ℓ ,s ,s ) 2 2 2 : 1 2 1 2 1 2 ∈ I ×L ×S (3) n u +s u +s 1 1 2 2 (mod p) ℓ ≡ ℓ 1 2 o for some special class of sets. We say that a set F is h-spaced if no elements s ,s and p 1 2 S ⊆ ∈ S positive integer k h satisfy the equality s +k = s . 1 2 ≤ The following result is given in [11] and is based on some ideas of Shao [27]. Lemma 13. If L < H and 2HL < p then for any H-spaced set for S W, given by (3) we have (# HL)2 W S +# HLpo(1). ≪ p S We also define (4) U = U(v)2, vX∈Fp where u+f(λ) (5) U(v) = # (u,ℓ,λ) : v (mod p) . ∈ I ×L×G ℓ ≡ (cid:26) (cid:27) Lemma 14. For every fixed real ε > 0 and integer d 1 there are ≥ some δ > and η > 0 such that if T > pε and p1/2−ε H L ≥ ≥ then for U, given by (4) we have U HLT2p−η. ≪ 8 M.-C. CHANGANDIGOR E. SHPARLINSKI Proof. Let be the largest H-separated subset of = f(λ) : λ 1 0 S F { ∈ . By Lemma 10 we have # pκ for some fixed κ > 0. 1 G} S ≫ Inductively, we define as the largest H-separated subset of k+1 S k = , k = 1,2,.... k k−1 j F F \ S j=1 [ Clearly for some b F and a set = b+1,...,b+H we have p ∈ J { } # k #( ) F . k F ∩J ≥ # k+1 F On the other hand, by Lemma 10 #( ) #( ) Tp−κ. k 1 F ∩J ≤ F ∩J ≪ Hence there is a partition K = 0 k F S k=0 [ into disjoined sets with K Tp−κ/2 such that ≤ # Tp−κ/2, 0 • S ≤ is H-separated with # pκ/2, k = 1,...,K. k k • S S ≥ For k = 0,...,K we define u+s U (v) = # (u,ℓ,s) : v (mod p) . k k ∈ I ×L×S ℓ ≡ (cid:26) (cid:27) We have K K U(v) = U (v) = U (v)+ U (v). k 0 k k=0 k=1 X X So, squaring out and summing over all v F , we obtain p ∈ 2 K U U (v)2 + U (v) 0 k ≪ ! vX∈Fp vX∈Fp Xk=1 K = U (v)2 + U (v)U (v). 0 k m vX∈Fp vX∈FpkX,m=1 Now, changing the order of summation in the second term in the above and then using the Cauchy inequality, yields (6) U V +V2, ≪ 1 2 DOUBLE CHARACTER SUMS OVER SUBGROUPS AND INTERVALS 9 where 1/2 K V = U (v)2 and V = U (v)2 . 1 0 2 k   vX∈Fp Xk=1 vX∈Fp   We have, V = # (u ,u ,ℓ ,ℓ ,s ,s ) 2 2 2 : 1 1 2 1 2 1 2 ∈ I ×L ×S0 n u +s u +s 1 1 2 2 (mod p) ℓ ≡ ℓ 1 2 o max # (u ,u ,ℓ ,ℓ ) 2 2 : 1 2 1 2 ≤ s1,s2∈Fp ∈ I ×L n u +s u +s 1 1 2 2 (mod p) . ℓ ≡ ℓ 1 2 o Since L H p1/2−ε, by Lemma 12 we obtain ≤ ≤ (7) V (# )2HL(logp)2 HLT2p−ε(logp)2. 1 0 ≪ S ≪ Furthermore, Lemma 13 implies that for k = 1,...,K we have U (v)2 (# HL)2p−1 +# HLpo(1). k k k ≪ S S vX∈Fp Hence, applying the Cauchy inequality, we derive K V # HLp−1/2 +(# )1/2H1/2L1/2po(1) 2 k k ≪ S S k=1 X(cid:0) (cid:1) K HLTp−1/2 +H1/2L1/2po(1) (# )1/2 k ≤ S k=1 X 1/2 K HLTp−1/2 +H1/2L1/2po(1) K # k ≤ S ! k=1 X HLTp−1/2 +H1/2K1/2L1/2T1/2po(1). ≤ Since K Tp−κ/2 and L H p1/3, we see that ≤ ≤ ≤ (8) V HLTp−1/2 +H1/2L1/2Tp−κ/2+o(1) H1/2L1/2Tp−κ/2+o(1) 2 ≪ ≤ (assuming that κ is small enough). Substituting (7) and (8) in (6), leads us to the bound U HLT2p−εlogp+HLT2p−κ+o(1) ≪ and the result follows. ⊔⊓ 10 M.-C. CHANGANDIGOR E. SHPARLINSKI Let E( ) betheadditive energy ofamultiplicative subgroup F∗, G G ⊆ p that is E( ) = # (λ ,µ ,λ ,µ ) 4 : λ +µ = λ +µ . 1 1 2 2 1 1 2 2 G { ∈ G } By a result of Heath-Brown and Konyagin [19], if # = T p2/3 we G ≤ have E( ) T5/2. G ≪ Recently, Shkredov [28] has given an improvement which we present in the following slightly less precise form (which supreses logarithmic factors in po(1)). Lemma 15. For T p2/3 we have ≤ T32/13po(1), if T p1/2, ≤ E( ) T31/13p1/26+o(1), if p1/2 < T p29/48, G ≤  ≤ T3p−1/3+o(1), if p29/48 < T p2/3.  ≤ .  3. Proofs of main results 3.1. Proof of Theorem 1. We have 1 (9) S (a, , ) = W, χ I G T where W = χ(µ)χ(µx+aλ). x∈I λ,µ∈G X X (since χ(µ) = χ(µ−1) for µ F∗). Hence ∈ p W χ(xµ+aλ) . | | ≤ (cid:12) (cid:12) Xx∈I λX,µ∈G(cid:12)Xλ∈G (cid:12) (cid:12) (cid:12) Collecting the products µx with th(cid:12)e same value u(cid:12) F , we obtain (cid:12) (cid:12)∈ p W R(u) χ(u+aλ) , | | ≤ (cid:12) (cid:12) uX∈Fp (cid:12)Xλ∈G (cid:12) (cid:12) (cid:12) where (cid:12) (cid:12) (cid:12) (cid:12) R(u) = # (x,µ) : µx = u . { ∈ I ×G } So, by the Cauchy inequality, 2 W 2 R(u)2 χ(u+aλ) . | | ≤ (cid:12) (cid:12) uX∈Fp uX∈Fp(cid:12)Xλ∈G (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)

Double Character Sums over Subgroups and Intervals

Similar Double Character Sums over Subgroups and Intervals