THE ERDÖS-FALCONER DISTANCE PROBLEM, EXPONENTIAL SUMS, AND FOURIER ANALYTIC APPROACH TO INCIDENCE THEOREMS IN VECTOR SPACES OVER FINITE FIELDS

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1 THE ERDÖS-FALCONER DISTANCE PROBLEM, EXPONENTIAL SUMS, AND FOURIER ANALYTIC APPROACH TO INCIDENCE THEOREMS IN VECTOR SPACES OVER FINITE FIELDS ALEX IOSEVICH AND DOOWON KOH Abstact. We study the Edös/Falcone distance poblem in vecto spaces ove finite fields with espect to the cubic metic. Estimates fo discete Aiy sums and Adolphson/Spebe estimates fo exponential sums in tems of Newton polyheda play a cucial ole. Simila techniues ae used to study the incidence poblem between points and cubic and uadatic cuves. As a esult we obtain a non-tivial ange of exponents that appea to be difficult to attain using combinatoial methods. Key wods. distance sets, Newton diagams, Gauss sums, multiplicative chaactes, exponential sums, Kloosteman sums. AMS subject classifications. 52C10 1. Intoduction The Edös distance poblem. The Edös distance conjectue in the Euclidean space says that if E is a finite subset of R d, d 2, then 1.1) # E) #E) 2 d, whee E) = { x y : x,y E}, with x y 2 = x 1 y 1 ) x d y d ) 2 and hee, and thoughout the pape, X Y means that thee exists C > 0 such that X CY, and X Y, with the contolling paamete N, means that fo evey ǫ > 0 thee exists C ǫ > 0 such that X C ǫ N ǫ Y. Taking E = Z d [0,N 1 d ] d shows that 1.1) cannot in geneal be impoved. The conjectue has not been solved in any dimension. See, fo example, [14], [1], and the efeences contained theein fo the desciption of the conjectue, backgound mateial, and a suvey of ecent esults. In this pape we study the Edös distance poblem in vecto spaces ove finite fields. This poblem was ecently addessed by Tao [19]) who elates it to some inteesting uestions in combinatoics, and, moe ecently, by Iosevich and Rudnev. We shall descibe these esults late in the intoduction. Let F denote the finite field with elements, and let F d denote the d-dimensional vecto space ove this field. Let E F d, d 2. Then a possible analog of the classical Edös distance poblem is to detemine the smallest possible cadinality of the set n E) = { x y n = x 1 y 1 ) n x d y d ) n : x,y E}, with n a positive intege 2, viewed as a subset of F. This wok was suppoted by the NSF Gant DMS Mathematics Depatment, 202 Mathematical Sciences Bldg, Univesity of Missoui, Columbia, MO USA iosevich@math.missoui.edu). Mathematics Depatment, 202 Mathematical Sciences Bldg, Univesity of Missoui, Columbia, MO USA koh@math.missoui.edu). 1

2 2 A. IOSEVICH AND D. KOH In the finite field setting, the estimate 1.1) cannot hold without futhe estictions. To see this, let E = F d. Then #E = d and # E) =. Futhemoe, an inteesting featue of the Edös distance poblem in the finite field setting with n = 2 is the existence of non-tivial sphees of zeo adius. These ae sets of the fom {x F d : x 2 1 +x x 2 d = 0} and seveal assumptions in the statements of esults below ae thee pecisely to deal with issues ceated by the pesence of this object. Fo example, suppose 1 is a suae in F. Using sphees of zeo adius one can show, in even dimensions, that thee exists a set of cadinality pecisely d 2 such that all the distances, x 1 y 1 ) x d y d ) 2 ae zeo. What s moe, suppose F is a finite field, such that = p 2, whee p is a pime. Then E = F d p is natually embedded in F d, has cadinality d 2, and detemines only distances. If n > 2, the situation is eually fascinating. Fo example, if n = 3 and d = 2, the euation x 3 1 +x 3 2 = 0 always has at least solutions, since cube oot of 1 is 1. This euation may have as many as 3 solutions if the pimitive cube oot of 1 is in the field. With these examples as guide, we genealize the conjectue oiginally stated in [10] in the case n = 2 as follows. Conjectue 1.1. Let E F d of cadinality C d 2, with C sufficiently lage. Then # n E). The authos conjectue in [10] that the constant C that appeas above may be taken to be any numbe bigge than one, at least in the case n = 2. It is inteesting to note that if n > 2, the situation becomes moe complicated. Fo example, as we pointed out above, if n = 3 and d = 2, the numbe of points on the cuve x x 3 2 = 0 may be as high as 3, depending on whethe o not the pimitive cube oot of 1 is in the field. Thus a coesponding conjectue in the case n > 2 must be designed with these issues in mind. 2. Pevious esults. A Euclidean plane agument due to Edös [6]) can be applied to the finite field set-up unde the assumption of Conjectue 1.1 to show that if d = 2 and #E C, with C sufficiently lage, then 2.1) # n E) #E) 1 2. This esult was impoved by Bougain, Katz and Tao [3]) who showed using inticate incidence geomety that fo evey ǫ > 0, thee exists δ > 0, such that if #E 2 ǫ, then # 2 E) 1 2 +δ. The elationship between ǫ and δ in the above agument is difficult to detemine. Moeove, mattes ae even moe subtle in highe dimensions in the context of vecto spaces ove finite fields, because intesection of analogs of sphees, both uadatic and cubic, in F d may be uite complicated, and the standad induction on the dimension agument in R d see e.g. [1]) that allows one to bootstap the estimate 2.1) into the estimate 2.2) # R de) #E) 1 d

3 THE ERDÖS-FALCONER DISTANCE PROBLEM 3 does not immediately go though. We establish the finite field analog of the estimate 2.2) below using Fouie analytic methods and numbe theoetic popeties of Kloosteman sums and its moe geneal analogs. Anothe way of thinking of Conjectue 1.1 is in tems of the Falcone distance conjectue, [7], in the Euclidean setting which says that if the Hausdoff dimension of a set in R d exceeds d 2, then the Lebesgue measue of the distance set is positive. Conjectue 1.1 implies that if the size of the set is geate than d 2, then the distance set contains a positive popotion of all the possible distances, an analogous statement. In [10], the authos poved the following esult. Theoem 2.1. Let E F d, d 2, such that #E C d+1 2. Then if C is sufficiently lage, 2 E) contains evey element of F. 3. Main esults of this pape Distances detemined by a single set. Ou fist esult is the vesion of Theoem 2.1 fo cubic metics. Theoem 3.1. Suppose that is a pime numbe conguent to 1 modulo 3. Let E F d, such that #E C d+1 2. Then if C is sufficiently lage, 3 E) contains evey element of F. Suppose that d = 2, and n 2. Then if #E C 3 2 fo C sufficiently lage, then n E) contains evey elements of F. Coollay 3.2. Suppose that is a pime numbe conguent to 1 modulo 3. Let E F d, d 2, such that #E = C d+1 2. Then if C is sufficiently lage, # 3 E) #E) 2 d+1. In two dimensions, the same conclusion, with d = 2, holds fo any n 2. Note that in the case d = 2, the exponent 2 3 obtained via the coollay, fo the given ange of paametes, is a much bette exponent than the one obtained by the incidence agument due to Edös descibed in 2.1) above. Also, we point out once moe that Edös agument does not genealize to highe dimensions, at least not vey easily, due to the possibly complicated intesection popeties of cubic vaieties Szemeédi-Totte type Incidence theoems and distances between pais of sets. As in the case n = 2, the poof of Theoem 3.1 can be modified to yield a good uppe bound on the numbe of incidences between points and cubic sufaces in vecto spaces ove finite fields. It is an analog, and a highe dimensional genealization, of the following classical esult due to Szemeédi and Totte. Theoem 3.3. The numbe of incidences between N points and M lines o cicles of the same adius) in the plane is N + M + NM) 2 3. Ou incident estimate is the following.

4 4 A. IOSEVICH AND D. KOH Theoem 3.4. Suppose that is a pime numbe conguent to 1 modulo 3. Let E,F F d, d 2. Then if j 0, #{x,y) E F : x 1 y 1 ) x d y d ) 3 = j} #E #F 1 + d 1 2 #E) 12 #F) 1 2. Similaly, if is a pime numbe and j 0, then #{x,y) E F : x 1 y 1 ) x d y d ) 2 = j} #E #F 1 + d 1 2 #E) 12 #F) 1 2. In two dimensions, the same esult holds, with d = 2, with 3 eplaced by n fo any n 2. Remak 3.5. In paticula, if #E #F d+1 2, then the numbe of incidences between points in E and sphees, uadatic o cubic, centeed at elements of F is d. To make the numeology moe tanspaent, Theoem 3.4 says that if N d+1 2, the numbe of incidences between N points and N sphees, cubic o uadatic, in F d is d = N 2d d+1. In two dimensions this says that the numbe of incidences between N points and N cicles is N 4 3, povided that N d+1 2, matching in this setting the exponent in the celebated esult due to Szemeédi and Totte in the Euclidean plane see Theoem 3.3 above). An easy modification of the method used to pove Theoem 3.4 above yields the following distance set esult. Coollay 3.6. Let E,F F d, d 2. Suppose that is a pime numbe conguent to 1 modulo 3 and #E #F C d+1. Let 3 E,F) = { x y 3 : x E,y F }. Then if C is sufficiently lage, then 3 E,F) contains evey element of F. As befoe, in two dimensions the same conclusion holds, with d = 2, with 3 eplaced by n E). Obseve that if E = F, then we can safely say that in fact 3 E,F) contains evey element of F, but if E F, the zeo distance may not be pesent. We also call the eade s attention to the fact that an analogous vesion of this esult was independently obtained by Shpalinski in [17]. 4. Fouie analytic peliminaies and notation. Let F be a finite field with elements, whee is a pime numbe. Let χt) = e 2πi t. Given a complex valued function f on F d, define the Fouie tansfom of f by the euation fm) = d x F d χ x m)fx).

5 THE ERDÖS-FALCONER DISTANCE PROBLEM 5 We also need the following basic identity, typically known as the Plancheel theoem. Let f be as above. Then m F d fm) 2 = d x F d fx) Poof of the fist pat of Theoem 3.1. Let χs) = e 2πi s. Let S j denote the chaacteistic function of the cubic sphee whee, as above, {x F d : x 3 = j}, x 3 = x x 3 d. The key estimate of the pape is the following. Theoem 5.1. Let x 3 = x x 3 d. Suppose that is a pime numbe conguent to 1 modulo 3 and j 0. Then if m 0,...,0), then Ŝjm) = and if m = 0,...,0), then d χx m) {x F d : x 3=j} Ŝ j m) = 1 + O d+1 2 ) 1. d+1 2, Fo j 0, conside #{x,y) E E : x y 3 = j} = Ex)Ey)S j x y) x,y F d = 2d m Êm) 2 Ŝ j m) = A + B, whee A = 2d Ê0,...,0) 2 Ŝ j 0,...,0), and B = 2d Êm) 2 Ŝ j m). m 0,...,0) Using the second pat of Theoem 5.1, A 2d 2d #E) 2 1.

6 6 A. IOSEVICH AND D. KOH Wheeas using the fist pat of Theoem 5.1, B 2d d+1 2 m 0,,0) Êm) 2 whee and 2d d+1 We theefoe obtain that 2 d x F d E 2 x) = d 1 2 #E. #{x,y) E E : x y 3 = j} = A + B, A #E) 2 1, B #E d 1 2. We conclude that if #E C d+1 2, with C sufficiently lage, then #{x,y) E E : x y 3 = j} > 0 fo each j 0. This completes the poof of Theoem Poof of Theoem 5.1. We have Ŝ j m) = d {x F d : x 3=j} χ x m) = 1 δm) + d 1 x χt x 3 j))χ x m), t F whee δm) = 1 if m = 0,...,0) and 0 othewise. Lemma 6.1. Let χ be a nontivial additive chaacte of F with conguent to 1 modulo 3. Suppose that m = m 1,,m l ) F ) l. Then fo any multiplicative chaacte ψ of F of ode 3 and t 0, we have = ψ l t) s 1,,s l F l j=1 s j F χ s j m j + s 3 jt) χs s l + m 3 1t 1 s m 3 l t 1 s 1 l )ψs 1 ) ψs l ), whee 3 3 m 3 j is denoted by m3 j in the ight-hand side of the euation. We shall also need the following esult due to Duke and Iwaniec [4]).

7 THE ERDÖS-FALCONER DISTANCE PROBLEM 7 Theoem 6.2. Suppose that is conguent to 1 modulo 3 and let ψ be a multiplicative chaacte of ode thee. Then fo any a F. It follows that χas 3 + s) = ψsa 1 )χs 3 3 as) 1 ), s F s F χ sm j + s 3 t) = χs s 3 tm 3 j ) s F s F = s F ψst 1 )χs + m 3 jt s 1 ). since ψ is a multiplicative chaacte of F of ode thee and m j 0. Absobing 3 3 into m j to make the notations simple, we complete the poof of Lemma 6.1. Lemma 6.3. Let χ be a nontivial additive chaacte of F with conguent to 1 modulo 3. Then fo any multiplicative chaacte ψ of F of ode 3 and t 0, we have ) l l χts 3 l ) = s F =0 ) l ψ l+) t) ψ 1) ) l ψ2 1)), ). whee is a binomial coefficient, l is a positive intege, and the Fouie tansfom of a multiplicative chaacte ψ of F is given. by ψv) = 1 s F χ vs)ψs). Remak 6.4. ψv) = O 1 2 ) fo v 0. To pove Lemma 6.3, we need the following theoem. Fo the poof, see the [13], page 217, Theoem 5.30). Theoem 6.5. Let χ be a nontivial additive chaacte of F, n N, and ψ a multiplicative chaacte of F of ode h =gcdn, 1). Then h 1 χts n + b) = χb) ψ k t)gψ k,χ) s F fo any t,b F with t 0, whee Gψ k,χ) = s F ψ k s)χs). By using Theoem 6.5, we see that fo any multiplicative chaacte ψ of ode thee, ) l 2 χts 3 ) = s F k=1 k=1ψ k t) s F ) l ψ k s)χs)

8 8 A. IOSEVICH AND D. KOH = ψ 1 t) s F ψs)χs) + ψ 2 t) s F ) l ψ 2 s)χs) whee and = G 1 t) + G 2 t) ) l = l =0 l G 1 t) = ψ 1 t) s F G 2 t) = ψ 2 t) s F ) G 1 t) l G 2 t), ψs)χs) ψ 2 s)χs). Note that G 1 t) = ψ 1 t) ψ 1) and G 2 t) = ψ 2 t) ψ 2 1). Thus we conclude that ) l l χts 3 l ) = s F =0 ) l ψ l+) t) ψ 1) ) l ψ2 1)). We ae now eady to pove Theoem 5.1. Fist, we assume that m = 0,,0) F d. Then, using Lemma 6.3, we see that Ŝ j 0,,0) = d {x F d : x 3=j} 1 = 1 + d 1 t F χ tj) x χt x 3 )) = 1 + d 1 t F χ tj) d =0 d ) ) d ) d ψ t) d+) ψ 1) ψ2 1) = d =0 d ) ) d ) ψ 1) ψ2 1) t F χ tj)ψ d+) t) = d =0 d ) ) d ) ψ 1) ψ2 1) ψ d+) j) = 1 + O d+1 2 ) 1.

9 THE ERDÖS-FALCONER DISTANCE PROBLEM 9 In the last euality, we used the fact that ψv) = O 1 2 ) fo any multiplicative chaacte of F with v 0. Thus the second pat of Theoem 5.1 is poved. In ode to pove the fist pat of Theoem 5.1, we shall deal with the poblem in case m = m 1,,m d ) 0,,0). Suppose that m j 0 fo j J {1,2,,d} and m j = 0 fo j {1,2,,d} \ J = J. Without loss of geneality, we may assume that J = {1,2,,l} and J = {l + 1,,d} fo some l = 1,2,,d. Using Lemma 6.1 and Lemma 6.3, we see that Ŝ j m) = d 1 t F χ tj) x F d χt x 3 m x) = d 1 t F = d 1 t F χ tj)ψ l t) l χ tj) k=1 s 1,,s l F ) d χts 3 k m k s k ) s k F k=l+1 ) χts 3 k) s k F χs 1 + +s l +m 3 1t 1 s m3 l t 1 s 1 l )ψs 1 ) ψs l ) d l d l =0 ) ) d l ) d l ψ t) d l+) ψ 1) ψ2 1) d l = 1 l d l =0 ) ) d l ) ψ 1) ψ2 1) t F χ tj)ψ d+) t) s 1,,s l F χs s l + m 3 1t 1 s m 3 l t 1 s 1 l )ψs 1 ) ψs l ). Since d l ) ) d l ) ψ 1) ψ2 1) = O 1 2 d l) ), we obtain that whee A χ,ψ) is given by χ tj)ψ d+) t) t F Ŝjm) s 1,,s l F d+l 1 2 d l =0 A χ,ψ), χs 1 + +s l +m 3 1t 1 s m3 l t 1 s 1 l )ψs 1 ) ψs l ). We now apply the esult of Adolphson and Spebe [2], Theoem 4.2, Coollay 4.3) to see that fo all = 0,1,,d l, This completes the poof. A χ,ψ) l+1 2.

10 10 A. IOSEVICH AND D. KOH 7. Poof of the second pat of Theoem 3.1. As in the poof of the fist pat of Theoem 3.1, it suffices to pove the following estimation. Theoem 7.1. Let x n = x n 1 + x n 2 fo x F 2 and n 2. Suppose that is a pime numbe and j 0. Then if m 0,0), then and if m = 0,0),then Ŝjm) = 2 χ x m) {x F 2 : x n=j} 3 Ŝ j m) = 1 + O 3 2 ) 1. 2, To pove Theoem 7.1, we obseve that fo j 0 and m F 2, Ŝ j m) = 2 χ x m) {x F 2 : x n=j} = 1 δm) + 3 x χt x n j))χ x m), t F whee δm) = 1 if m = 0,0) and 0 othewise. Fist we shall pove the second pat of Theoem 7.1. Let ψ be a multiplicative chaacte of F of ode h = gcdn, 1). Fo each i = 1,2,...,h 1), we denote by β i a non-negative intege. Then by Theoem 6.5, we see that ) 2 χts n ) s F = β 1+ +β h 1 =2 It theefoe follows that Ŝ j 0,0) = 1 + 2! ψ β1+ +h 1)βh 1) t) 2 ) β1 ) βh 1. ψ 1) ψh 1 1) β 1! β h 1! β 1+ +β h 1 =2 2! ) β1 ψ j) β 1! β h 1! γh,β) ψ 1) ψh 1 1) ) βh 1 whee γh,β) is given by β 1 + 2β h 1)β h 1. Since ψv) = O 1 2 ) fo each multiplicative chaacte ψ and v F, we conclude Ŝ0,0) = 1 + O 3 2 ) 1. This completes the poof of the second pat of Theoem 7.1. It emains to pove the fist pat of Theoem 7.1. The cohomological intepetation can be used to estimate the exponential sums. We now intoduce the cohomology theoy based on wok of authos in [5] and [2]. Let g be a polynomial given by 7.1) g = α J A α x α F [x 1,,x d ],

11 THE ERDÖS-FALCONER DISTANCE PROBLEM 11 whee J is a finite subset of N {0}) d, and A α 0 if α J. We denote by g) the Newton polyhedon of g which is the convex hull in R d of the set J 0,,0). Fo any face σ of any dimension) of g), we put g σ = A α x α. α σ J Definition 7.2. Let g F [x 1,,x d ] be a polynomial as in 7.1). We say that g is nondegeneate with espect to g) if fo evey face σ of g) that does not contain the oigin, the polynomials g σ x 1,, g σ x d ) d have no common zeo in F whee F denotes an algebaic closue of F. We say that g is commode with espect to g) if fo each k = 1,2,,d, g contains a tem A k x α k k fo some α k > 0 and A k 0. The geneal vesion of the following theoem can be found in [5] see Theoem 9.2). Theoem 7.3. Let be a pime numbe. Suppose that g : F d F,d 2, is commode and nondegeneate with espect to g). Then χgx)) = O d 2 ). x F d We now pove the fist pat of Theoem 7.1. Since m 0,0), we have x F χ x 2 m) = 0. We theefoe see that fo j 0, Ŝ j m) = 3 χgt,x 1,x 2 )) = 3 χgt,x 1,x 2 )), t,x 1,x 2) F F2 t,x 1,x 2) F 3 whee gt,x 1,x 2 ) = tx n 1 + tx n 2 m 1 x 1 m 2 x 2 jt. If m 1 m 2 0, then g is commode. By Theoem 7.3, it suffices to show that g is nondegeneate with espect to g). Note that g) has five zeo-dimensional faces, eight one-dimensional faces and thee two-dimensional faces which do not contain the oigin. It is easy to show that fo evey face σ of g) that does not contain the oigin, the polynomials g σ t, g σ x 1, g σ x 2 have no common zeo in F ) 3 because we may assume that is sufficiently lage and so n is not conguent to 0 modulo. This implies that g is nondegeneate with espect to g). We now assume that m 1 m 2 = 0. Without loss of geneality, we may assume that m 1 0, and m 2 = 0 because m 0,0). By using Theoem 6.5, we obtain that fo a multiplicative chaacte ψ of F of ode h = gcdn, 1), h 1 Ŝ j m) = 3 χtx n 1 m 1 x 1 jt) ψ k t) ψ k 1) t,x 1) F F k=1

12 12 A. IOSEVICH AND D. KOH h 1 = 2 k=1 ψ k 1) t,x 1) F F ψ k t)χtx n 1 m 1 x 1 jt) whee R k ψ k,χ) is given by h R k ψ k,χ), k=1 t,x 1) F F ψ k t)χtx n 1 m 1 x 1 jt). Fo each k = 1,2,,h 1, define ψ k 0) = 0. Then we can obtain that R k ψ k,χ) = ψ k t)χtx n 1 m 1 x 1 jt). t,x 1) F F Applying Theoem 7.3, we have This completes the poof. R k ψ k,χ) = O). 8. Poof of Theoem 3.4 and Coollay 3.6. As we mentioned in the intoduction, this is a simple vaiation on the poof of Theoem 3.1. Indeed, #{x,y) E F : x y n = j} = 2d m Êm) Fm)Ŝjm) = #E #F Ŝj0,,0) + 2d m 0,...,0) By the second pat of Theoem 5.1o Theoem 7.1), I #E #F 1. Êm) Fm)Ŝjm) = I + II. Applying Cauchy-Schwatz, Theoem 5.1 o Theoem 7.1) and Plancheel, we see that II 2d d+1 2 Êm) Fm) m 0,...,0) 2d d+1 2 2d d+1 2 d m ) 1 ) 1 2 Êm) 2 Fm) 2 2 x m ) 1 ) Ex) 2 Fx) 2 x

13 THE ERDÖS-FALCONER DISTANCE PROBLEM 13 This completes the poof of Theoem 3.4. = d 1 2 #E #F. In ode to pove Coollay 3.6, we obseve that by the second pat of Theoem 5.1o Theoem 7.1), I #E #F 1. On the othe hand, we have seen above that II d 1 2 #E #F, and the esult follows by a diect compaison. REFERENCES [1] P. Agawal and J. Pach, Combinatoial geomety, Wiley-Intescience Seies in Discete Mathematics and Optimization. A Wiley-Intescience Publication. John Wiley and Sons, Inc., New Yok 1995). [2] A. Adolphson and S. Spebe, Exponential sums and Newton polyheda - cohomology and estimates, Ann. Math., ), pp [3] J. Bougain, N. Katz and T. Tao, A sum-poduct estimate in finite fields, and applications, Geom. Funct. Anal., ), pp [4] W. Duke and H. Iwaniec, A elation between cubic exponential and Kloosteman sums, Contemp. Math., ), pp [5] J. Denef, and F. Loese,Weights of exponential sums, intesection cohomology, and Newton polyheda, Invent. Math., ), pp [6] P. Edös, On sets of distances of n points, Ame. Math. Monthly., ), pp [7] K. J. Falcone,On the Hausdoff dimensions of distance sets, Mathematika., ), pp [8] B. J. Geen, Restiction and Kakeya phenomena, Lectue notes 2003) bjg23/kp.html). [9] H. Iwaniec and E. Kowalski, Analytic Numbe Theoy, Collouium Publications., ). [10] A. Iosevich and M. Rudnev, Edös/Falcone distance poblem in vecto spaces ove finite fields, Tans. Ame. Math. Soc., ), no. 12, pp electonic). [11] N. Katz, Gauss sums, Kloosteman sums, and monodomy goups, Ann. Math. Studies., 116, Pinceton 1988). [12] M. Lacey and W. McClain, On an agument of Shkedov in the finite field setting, 2006), On-line jounal of analytic combinatoics [13] R. Lidl and H. Niedeeite, Finite fields, Cambidge Univ. Pess 1997). [14] J. Matoušek, Lectues on Discete Geomety, Gaduate Texts in Mathematics, Spinge ). [15] G. Mockenhaupt and T. Tao, Restiction and Kakeya phenomena fo finite fields, Duke Math. J., ), pp [16] H. Niedeeite, The distibution of values of Kloosteman sums, Ach. Math., ), pp [17] I. Shpalinski, On the set of distances between two sets in vecto spaces ove finite fields, 2006), pepint). [18] E. Stein and R. Shakachi, Fouie analysis, Pinceton Lectues in Analysis, 2003). [19] T. Tao, Finite field analogues of Edös, Falcone, and Fustenbeg poblems, pepint. [20] A. Weil, On some exponential sums, Poc. Nat. Acad. Sci. U.S.A ), pp