An Estimate of Incomplete Mixed Character Sums 1 2. Mei-Chu Chang 3. Dedicated to Endre Szemerédi for his 70th birthday.

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1 An Estimate of Incomlete Mixed Chaacte Sums 2 Mei-Chu Chang 3 Dedicated to Ende Szemeédi fo his 70th bithday. 4 In this note we conside incomlete mixed chaacte sums ove a finite field F n of the fom x B H ψ ( f(x) ) χ(x), whee ψ is an additive chaacte, f(x) F n a olynomial, χ a non-tivial multilicative chaacte and B H a box of the fom B H = { n j= x jω j : x j [, H]}. (Hee {ω i } n i= is an abitay basis of F n ove F.) If f(x) = 0 and n =, Bugess well-known theoem ovides a nontivial estimate unde the assumtion H > /4+. A genealization to abitay finite fields was obtained in [C], [C2] and vey ecently [K], eventually oviding a statement of the same stength as Bugess, in F n. If n = and f(x) is linea, [FI] oved a non-tivial bound assuming H > /4+. Fo a geneal olynomial f(x) the only available esult ae that of P. Enflo [E] and a comment made by Heath-Bown [H] in the eview of [E]. Heath-bown s estimate (fo n = ) assumes again that H > /4+ and comes with a saving of the fom c()/2d, whee d is the degee of f(x). Ou esult below teats the situation of a field F n (elying on Konyagin s bound fo the multilicative enegy of a box B H as descibed above) and a olynomial f(x) of abitay degee d, assuming H > /4+. We obtain a saving ove the tivial bound of the fom c(n,)/(d+)2, so that, inteestingly, even fo n = the esult seems new. Notation and Convention.. e(θ) = e 2πiθ, e (θ) = e( θ ) 2. When thee is no ambiguity, = [ ] Z Mathematics Subject Classification.Pimay L40, L26; Seconday A07, B75. 2 Key wods. chaacte sums, quadatic esidues, Bugess 3 Reseach atially financed by the National Science Foundation. 4 Hay Bithday!

2 2 3. Multilicative enegy E(A, B) = { } (a, a 2, b, b 2 ) A A B B : a b = a 2 b 2. Let ω,..., ω n be an abitay basis fo F n ove F. Then fo any x F n, thee is a unique eesentation of x in tems of the basis. x = x ω + + x n ω n. A box B H F n of size H is a set such that fo each j, the coefficients x j fom an inteval. { n B H = x j ω j : x j [, H], j= } j. () Theoem. Let χ (esectively, ψ) be a non-incial multilicative (es. additive) chaacte of F n. Fo a basis ω, ω 2,..., ω n of F n ove F, let B H be a box as defined in () by the basis with H > 4 +κ fo some κ > 0. (2) Then fo a olynomial f F n of degee d, we have ψ ( f(x) ) χ(x) < c(n, κ)(d + ) 2 δ B, x B H whee δ = κ 2 n 4( + 2κ)(2n + (d + ) 2 ) and c(n, κ) is a constant deending on n and κ. Sketch of Poof. As in [C], [C2] and [K], we use Bugess method [Bu]. Let > 0 be secified late (see (6)) and let B 2 H be a box of size 2 H as defined in (). Fo y B 2 H and 0 < t <, since yt B H, we have ψ ( f(x) ) χ(x) ψ ( f(x + yt) ) χ(x + yt) x B H x B H B \ (B + yt) + (B + yt) \ B < 2n H n.

3 3 Hence ψ ( f(x) ) χ(x) x B H B 2 H x B H, y B 2 H 0<t< An additive chaacte is of this fom Exanding and we wite ψ ( f(x + yt) ) χ(x + yt) + O( H n ). ψ(z) = e (T ξz), fo some ξ F n. f(x + yt) = a d (x, y)t d + a d (x, y)t d + + a 0 (x, y), ψ ( f(x + yt) ) ( d = e T ξa j (x, y) (3) t j ). (4) Fix > 0 (to be secified late) and atition [0, ] d+ in boxes Q of size. Thee ae (d+) boxes. Patition B H B 2 H accoding to the boxes Q. whee Ω = { (x, y) B H B 2 H : B H B 2 H = Ω, ( ) T ξaj (x, y) j d+ } Q (mod ). Hence fo θ = (θ,,..., θ,d+ ) Q and (x, y) Ω, we have T ξa j (x, y) θ,j <, fo j =,..., d +. (5) Since t <, (4) and (5) imly that fo (x, y) Ω, ψ( f(x + yt) ) ( d ) e θ,j t j 2π j T ξa j (x, y) θ,j t j (6) < 2π(d + ) d,

4 4 fo = (d + ). (7) Theefoe, the bound in (3) is bounded by B 2 H (x,y) Ω ( d ) e θ,j t j χ(x+yt) +O( H n ). (8) Fo z F n, denote µ (z) = { (x, y) Ω : x y = z}. (9) The sum in the fist tem of (8) equals ( d ) µ (z) e θ,j t j χ(z + t). (0) z F n Take Z secified late. Hölde s inequality bounds (0) by ( z F n ) µ (z) } {{ } (A) ( Hölde s inequality also gives ( (A) µ (z) = (,z z F n ( d ) e θ,j t j χ(z + t) ) } {{ } (B) () ) (,z µ (z) 2 ) Ω ) E(BH, B 2 H) (2). c(n) ( 2 H 2) n( )( 2n H 2n ) log. Hee the equality follows fom the definitions of µ (z) and the multilicative enegy. Fo the last inequality, we use Konyagin s bound on multilicative enegy [K] and that E(B H, B 2 H) E ( ) B H, B 2 H E ( B 2 H, B 2 H ( ) This is by Cauchy-Schwaz. (See [TV] Coollay 2.0.) ) 2.

5 5 To bound (B), we wite (B) = B with B = z Fo fixed, we exand = e ( d θ,j t j) χ(z + t). e( d θ,j t j) χ(z + t) and obtain ( d ) e θ,j t j χ(z + t) ( ) (3) (z + t ) (z + t ) c (t,..., t )χ (z + t t,..., t + ) (z + t ) with c (t,..., t ) =. This gives (B) () (d+) () (d+) t,..., t < [ n + n 2 (The last inequality is given by Weil s estimate.) Theefoe, z ( (z + t ) (z + t ) χ (z + t + ) (z + t )) ] [ (B) < c (d+) n + n ]. + (4) Putting (0)-(2) and (4) togethe, we have the fist tem of (8) bounded by c(n) log ( 2 H 2) n( )( ) 2n H 2n ( 2 H) n c(n) log H n n (d+) + n ( )[ ( c(n) H n (d+) 2 +2n 2 H [ ] n n ) n ( )[ <c(n) H n (d+) 2 +2n n κ n (d+) ( H ( The last inequality is by ou assumtion (2). ) ]. ) ] n [ n n 4 + ] (5)

6 6 and Take = = κ n (d + ) 2 + 2n (6) (2κ + ) n ( )( = (d + ) 2 + 2n 2 + ). (7) κ Substituting (6) in the second facto of (5), we obtain κn. Ou choice of imlies that > (2κ+) n κn and hence + n <. Theefoe, 4 4 (5) is bounded by κ 2 n 4(+2κ)((d+) 2 +2n). c(n, κ)(d + ) 2 H n ( 4 + κ n ) < c(n, κ)(d + ) 2 H n ( The inequality is because < 2 ( (d + ) 2 + 2n )( ) ) 2 + by (7). κ Remak. One may estimate the quantity c (t,..., t ), t,..., t which essentially equals to e ( d θ j t j) dθ0 dθ d Π d+ and may be estimated using the classical Vinogadov s mean value theoem. This will lead to some futhe saving of δ that may be significant fo secific values of κ and d. In the context of ou theoem whee we focus on small κ and lage d, the imovement tuns out to be without inteest. Acknowledgement. The autho would like to thank efeee fo many helful comments. The autho would also like to thank Lih-Chung Wang fo technical suot. Refeences [Bu] D.A. Bugess, On chaacte sums and imitive oots, Poc LMS (3), 2, (962), [BC] J. Bougain, M.-C. Chang, On a multilinea chaacte sum of Bugess, Comtes endus - Mathematique, to aea. [C] M.-C. Chang, On a question of Davenot and Lewis and new chaacte sum bounds in finite fields, Duke Math. J. 45 (2008), No. 3, [C2], Bugess inequality in F 2, Geom. Funct. Anal, to aea.

7 [C3] [E], Chaacte Sums in Finite Fields, AMS Contemoay Mathematics Seies, to aea. P. Enflo, Some oblems in the inteface between numbe theoy, hamonic analysis and geomety of Euclidean sace, Fist Intenational Confeence in Abstact Algeba, Quaestiones Math. 8 (995), no. -3, [FI] J. Fiedlande, H. Iwaniec, Estimates fo chaacte sums, Poc. Ame. Math. Soc. 9, No 2, (993), [H] D.R. Heath-Bown, MR (96h:079). [K] S.V. Konyagin, Estimates of chaacte sums in finite fields, Matematicheskie Zametki, to aea, (in Russian). [TV] T. Tao, V. Vu Additive Combinatoics, Cambidge Univesity Pess, (2006). 7 Deatment Of Mathematics, Univesity Of Califonia, Riveside, CA addess: mcc@math.uc.edu