A spherical extension theorem and applications in positive characteristic
aa r X i v : . [ m a t h . C A ] A ug A SPHERICAL EXTENSION THEOREM AND APPLICATIONS IN POSITIVECHARACTERISTIC
DOOWON KOH AND THANG PHAMA
BSTRACT . In this paper, we prove an extension theorem for spheres of square radii in F dq , which improves a result obtained by Iosevich and Koh (2010). Our main tool is a newpoint-hyperplane incidence bound which will be derived via a cone restriction theorem.We also will study applications on distance problems.
1. I
NTRODUCTION
Let q be an odd prime power, and F q be a finite field of order q . Let F dq be the d -dimensional vector space over F q . We endow the space F dq with counting measure d c .We denote the dual space of F dq by F dq ∗ , and endow it with normalized counting measure dn . For any algebraic variety V in F dq ∗ , we will endow it with the normalized surfacemeasure d σ which is defined by the relation d σ ( x ) = q d | V | V ( x ) dn ( x ), where | V | denotesthe cardinality of V .Let χ be a nontrivial additive character of F q . For any function g : F dq → C , the Fouriertransform of g is defined by b g ( x ) : = X m ∈ F dq g ( m ) χ ( − x · m ).If f is complex-value function on the dual space, namely, f : F dq ∗ → C , the inverse Fouriertransform of f is defined by f ∨ ( m ) : = q d X x ∈ F dq ∗ f ( x ) χ ( m · x ).In addition, the inverse Fourier transform of the measure f d σ is defined by( f d σ ) ∨ ( m ) : = | V | X x ∈ V f ( x ) χ ( m · x ).Since there is an isomorphism between F dq and its dual space F dq ∗ , for the sake of simplic-ity, we will simply write F dq for F dq ∗ , and in this paper, the only differences between twospaces are corresponding measures. Mathematics Subject Classification.
Let P be the paraboloid in F dq defined by the equation x d = x +· · · + x d − . For β ∈ F q , let P β be a translate of P by β defined by x d + β = x + · · · + x d − . For j
0, let S j be the spherecentered at the origin of radius j , namely, S j = { ( x , . . ., x d ) ∈ F dq : x + x + · · · + x d = j } .Notice that the definiton of a radius in finite fields is different from that in the Euclideancase.In this paper, the variety V will be often considered as a sphere or the paraboloid P .The L p → L r extension problem for the variety V is to determine all ranges of 1 ≤ p , r ≤ ∞ such that the following inequality(1.1) || ( f d σ ) ∨ || L r ( F dq , dc ) ≤ C || f || L p ( V , d σ ) holds for all functions f on V , where the positive constant C does not depend on q . Byduality, the extension estimate (1.1) implies that one has k b g k L p ′ ( V , d σ ) ≤ C k g k L r ′ ( F dq , dc ) for all functions g on ( F dq , d c ), where 1/ r + r ′ = p + p ′ =
1. We will use thenotation R ∗ V ( p → r ) ≪ X ≪ Y means that there exists someabsolute constant C > X ≤ C Y ,and X ∼ Y means Y ≪ X ≪ Y .Necessary conditions for R ∗ V ( p → r ) ≪ V and thecardinality of an affine subspace H lying on V . Mockenhaupt and Tao [21] indicated thatif V ⊂ F dq with | V | ∼ q d − and V contains an affine subspace H with | H | = q k , then one has(1.2) r ≥ dd − r ≥ p ( d − k )( p − d − − k ).In recent years, there has been intensive progress in studying L → L r and L p → L extension estimates for spheres. More precisely, for the case of R ∗ S j (2 → r ), it is believedthat(1.3) R ∗ S j (2 → r ) ≪ ⇐⇒ r ≥ d + d − d ≥ R ∗ S j (2 → r ) ≪ ⇐⇒ r ≥ d + d for even dimensions d ≥ f = H and f = S j ,where H denotes a maximal affine subspace lying in S j . Moreover, it follows from [14,Lemma 1.13] that | H | = q d − for odd d ≥
3, and | H | = q d − for even d ≥ SPHERICAL EXTENSION THEOREM AND APPLICATIONS 3
For all d ≥
2, Iosevich and Koh [8] used Stein-Tomas argument to obtain an L → L r extension result, which matches the conjecture (1.3). They also solved the extensionConjecture (1.4) in two dimensions. For the case d ≥ R ∗ S j (2 → r ) ≪ r ≥ d + d − d = k + − R ∗ P ³ → d + d + ´ ≪
1. The best current estimate is R ∗ P ³ → d + d ´ ≪ L → L r extension result can be derived by using the additive energy of a set on aparaboloid, for example, see [16, 17, 18, 22]. However, such a connection is not known forspheres. Moreover, the spherical extension problem is harder than the paraboloid case,since the Fourier transform of non zero-radii spheres is reduced to the Kloosterman sumwhose explicit form is not known yet.If − d = k +
2, Iosevich, Lee, Shen, and the authors [11] proved thatthe Conjecture (1.4) holds for the sphere of zero radius. The main difference between thezero radius and non-zero radius spheres is that we can use the Gauss sum in the placeof the Kloosterman sum in the Fourier decay. In addition, the explicit form of the Gausssum is very well-known, for instance, see [19].Similarly, the L p → L extension problem for the sphere S j is to determine all ranges of p such that the following inequality(1.5) || ( f d σ ) ∨ || L ( F dq , dc ) ≤ C || f || L p ( S j , d σ ) holds for all functions f on S j , where the positive constant C does not depend on q .In odd dimensional spaces, over the last ten years, it has been believed in [9, Section 2]that the Stein-Tomas exponent toward L p → L can not be improved in general. Thiscomes from the fact that if q ≡ d is odd, and the sphere S j is of non-zero squareradius, then S j contains an affine subspace of dimension d − . We refer the interestedreader to [9] for more details.In a recent paper, the authors and Vinh [14] showed that when we study spheres of primi-tive radii, then the Stein-Tomas exponent toward L p → L can be considerably improved.More precisely, the following results have been obtained in [14]. Theorem 1.1 ([14]) . Let g be a primitive element in F q . If d = k + then we haveR ∗ S g µ d d − → ¶ ≪ DOOWON KOH AND THANG PHAM
Theorem 1.2 ([14]) . Let g be a primitive element in F q . Suppose d = k − and q ≡ . Then we have R ∗ S g µ d d − → ¶ ≪ P , we know from [21] that if d = k + k ∈ N or d = k − q ≡ R ∗ P ( p → ≪ d − d − ≤ p ≤ ∞ .These estimates are optimal. Therefore, under the same conditions on q and d , thespherical extension theorems are much better. Based on dimensions of affine subspaces,the following conjecture has been provided in [14]. Conjecture 1.3.
Let S j be the sphere with non-zero radius in F dq . The following statementshold. (1)
If d = k + , k ∈ N , and j is not square, then the bound R ∗ S j ¡ d + d + → ¢ ≪ givesthe sharp L p → L estimate. (2) If d = k − , k ∈ N , q ≡ , and j is not square, then the bound R ∗ S j ¡ d + d + → ¢ ≪ gives the sharp L p → L estimate. (3) If d = k − , k ∈ N , q ≡ and j is square, then the bound R ∗ S j ¡ d + d + → ¢ ≪ gives the sharp L p → L estimate. In the proofs of Theorems 1.1 and 1.2, the main tool in [14] is the first association schemegraph , which works for spheres of primitive radii. However, in the case (3) of the Con-jecture 1.3, namely, when the radius of the sphere is a square, we get nothing from thatmethod. The main purpose of this paper is to address that case. Our main tool is anew point-hyperplane incidence bound in F dq , which will be derived via a cone restrictiontheorem. Our main result is the following. Theorem 1.4.
Suppose that d = k − , k ∈ N , and q ≡ . Let S j be a sphere ofsquare radius j in F dq . Then, we haveR ∗ S j µ d d − → ¶ ≪ SPHERICAL EXTENSION THEOREM AND APPLICATIONS 5
Applications:
We now discuss an application of L p → L extension estimates on a dis-tance problem.Given two points x = ( x , . . . , x d ) and y = ( y , . . ., y d ) in F dq , the distance function betweenthem is defined by the following equation || x − y || = ( x − y ) + · · · + ( x d − y d ) .For A ⊂ F dq , we denote the set of distances determined by pairs of points in A by ∆ ( A ),namely, ∆ ( A ) : = { || x − y || : x , y ∈ A } .The Erd˝os-Falconer distance problem over finite fields asks for the smallest number N such that for any A ⊂ F dq with | A | ≥ C q N the distance set ∆ ( A ) covers the whole field F q or a positive proportion of all possible distances.In 2007, Iosevich and Rudnev [12] proved that, for any A ⊂ F dq with d ≥
2, if the size of A is at least 4 q ( d + , then ∆ ( A ) = F q . It was proved in [7] that the exponent d + is sharpin odd dimensional vector spaces in the sense that there exists a set A ⊂ F dq with d ≥ | A | = o ³ q d + ´ and | ∆ ( A ) | = o ( q ). In even dimensions, it has been conjecturedthat the right exponent should be d /2, which is in line with the Falconer distance con-jecture in the continuous setting [4], which is still wide open. In two dimensional vectorspaces over finite fields, the best current exponent is for arbitrary finite fields [1] and for prime fields [20].For A ⊂ F dq , we define ∆ ( A ) : = { || x + x + x || : x i ∈ A } .Covert, Koh, and Pi [3] studied the following variant of the Erd˝os-Falconer distance prob-lem: For A ⊂ F dq , how large does A need to be such that ∆ ( A ) covers the whole field F q orat least a positive proportion of all elements?The geometric meaning of the size of ∆ ( A ) is the number of spheres centered at theorigin one would need to cover all centroids of triangles determined by triples of pointsin A .It has been indicated in [3] that in odd dimensions, in order to obtain | ∆ ( A ) | ≫ q , onemust have | A | ≫ q d + , but in even dimensions, we can decrease the threshold q d + to q d + − ǫ d for some ǫ d = ǫ ( d ) >
0. More precisely, they proved the following theorem.
Theorem 1.5.
Let A be a set in F dq with d even. (1) Suppose that d = and | A | ≫ q , then we have | ∆ ( A ) | = { || x + x + x || : x , x , x ∈ A } ≫ q . DOOWON KOH AND THANG PHAM (2)
Suppose that d ≥ and | A | ≫ q d + − d − + ǫ for any ǫ > , then we have | ∆ ( A ) | = { || x + x + x || : x , x , x ∈ A } ≫ q .The most interesting aspect of this result is that they have made a connection betweenthe size of ∆ ( A ) and L extension estimates for spheres in F dq . We now take an advantageof the sharp L estimate for spheres in even dimensions to improve Theorem 1.5. Ourimprovement is as follows. Theorem 1.6.
Let A be a set in F dq with d even. (1) Suppose that d = and | A | ≫ q , then we have | ∆ ( A ) | = { || x + x + x || : x , x , x ∈ A } ≫ q .(2) Suppose that d ≥ and | A | ≫ q d + d − d − , then we have | ∆ ( A ) | = { || x + x + x || : x , x , x ∈ A } ≫ q .2. A NEW INCIDENCE THEOREM
Let P be a set of points in F dq and Π be a set of hyperplanes in F dq . Let I ( P , Π ) be thenumber of incidences between P and Π , i.e. I ( P , Π ) = { ( p , π ) ∈ P × Π : p ∈ π } .It is well-known that(2.1) ¯¯¯¯ I ( P , Π ) − | P || Π | q ¯¯¯¯ ≤ q d − | P | | Π | .A proof can be found in [23] or [6] in the language of block designs from 1980s.For incidence bounds of this type, the value | P || Π | q is understood as the expected numberor the main term, and the value q d − | P | | Π | is the error term. There are several ex-amples that show that the error term can not be improved. Let us have a brief discussionhere. Example 2.1.
Assume that q ≡ d is odd, there exist sets P and Π such that | P | = | Π | = q d − and I ( P , Π ) = q d − | P | | Π | . Indeed, it has been proved in [7] that ifeither d is even and q ≡ d − = k , then there exist d − vectors { v , . . ., v d − } in F d − q , which are linearly independent, and v i · v j = i , j . Let A = Span ³ v , . . ., v d − ´ ⊂ F d − q . Then we have | A | = q d − . Given λ ∈ F q \ { } , define P = A × { λ } and Π being the set ofhyperplanes defined by the equation a x +· · ·+ a d − x d − + λ x d = λ with ( a , . . . , a d − ) ∈ A .Since || a − b || = || a || = || b || = a , b ∈ A , we have a · b =
0. Hence, the numberof incidences between P and Π is | P || Π | = q d − | P | | Π | . SPHERICAL EXTENSION THEOREM AND APPLICATIONS 7
In the same argument of Example 2.1, we also have the following.
Example 2.2.
Assume that q ≡ d = k + k ∈ N . Then there exist sets P and Π with | P | = | Π | = q d − such that I ( P , Π ) = q d − | P | | Π | .The main purpose of this section is to provide an improvement of the estimate (2.1) in thecase the point set P is distributed in at most q − ǫ spheres or translates of the paraboloid P for some 0 < ǫ < Theorem 2.3.
Let P be a set of points in F dq and Π be a set of hyperplanes in F dq . Let t bethe minimum number of spheres of square radii (or translates of the paraboloid P) thatcover the set P . We assume in addition that q ≡ and d = k − with k ∈ N . Thenthe number of incidences between P and Π satisfies ¯¯¯¯ I ( P , Π ) − | P || Π | q ¯¯¯¯ ≪ t q d − | P | | Π | + t q d − | P | | Π | .Theorem 2.3 is most effective when t is bounded by a constant number. Corollary 2.4.
Let P be a set of points in F dq and Π be a set of hyperplanes in F dq with | P | = | Π | . Suppose that P can be covered by a constant number of spheres of square radii(or translates of the paraboloid P). We assume in addition that q ≡ and d = k − with k ∈ N . Then the number of incidences between P and Π satisfies ¯¯¯¯ I ( P , Π ) − | P || Π | q ¯¯¯¯ ≪ q d − | P | | Π | + q d − | P | | Π | .3. P ROOF OF T HEOREM
Lemma 3.1.
For n ∈ N , let C n be the cone in F nq defined byC n : = © m ∈ F nq : m n = m + · · · + m n − ª . Suppose that n ≡ and q ≡ , then, for any G ⊂ F nq , we have k b G k L ( C n , d σ ) ≪ | G | + | G | q n .We are ready to prove Theorem 2.3. Proof of Theorem 2.3.
We first prove with t =
1. We consider two following cases:
Case : Assume that P lies on a sphere centered at the origin of radius r = u for some u ∈ F q \ { } . We can assume the center of the sphere is the origin since the number ofincidences is invariant under translations. Notice that a hyperplane in Π , given by theequation a · x = a d + , can be identified with a vector ( a , a d + ) in F dq × F q = F d + q . DOOWON KOH AND THANG PHAM
Define P ′ : = © ( λ p , λ u ) : p ∈ P , λ ∈ F q ª ⊂ F d + q ,and Π ′ : = © s ( a , . . . , a d , − u − · a d + ) : s ∈ F ∗ q , a x + · · · + a d x d = a d + ∈ Π ª .It is clear that | P ′ | = q | P | and | Π ′ | = q | Π | . Note that P ′ is a set on the cone C d + .We have I ( P , Π ) = X p ∈ P ,( a , a d + ) ∈ Π : a · p = a d + a · p = a d + is equivalent with the equation ( p , u ) · ( a , − u − a d + ) =
0, whichimplies that ( λ p , λ u ) · ( a , − u − a d + ) = λ ∈ F q . Hence, I ( P , Π ) = q − X x ∈ P ′ ,( a , a d + ) ∈ Π : x · ( a , − u − a d + ) = χ , the incidence I ( P , Π ) becomes | P ′ || Π | q + q − X x ∈ P ′ ,( a , a d + ) ∈ Π X s χ ( x · s ( a , − u − a d + )).From the definition of Π ′ , we get I ( P , Π ) = | P || Π | q + q X x ∈ P ′ , y ∈ Π ′ χ ( x · y ).By the Cauchy-Schwarz inequality,(3.1) ¯¯¯¯ I ( P , Π ) − | P || Π | q ¯¯¯¯ ≤ q X x ∈ P ′ | c Π ′ ( x ) | ≤ q | P ′ | à X x ∈ C d + | c Π ′ ( x ) | ! .Using Lemma 3.1 with G = Π ′ and n = d +
1, we have X x ∈ C d + | c Π ′ ( x ) | ≪ q d · ³ | Π ′ | + q − d + | Π ′ | ´ .Substituting this estimate into (3.1) gives us ¯¯¯¯ I ( P , Π ) − | P || Π | q ¯¯¯¯ ≪ q d − | P | | Π | + q d − | P | | Π | . Case : Assume that P lies on P β , recall, a translate of P by β defined by x d + β = x + · · · + x d − . Without loss of generality, we may assume that β = P ′ : = © ( λ p , λ ) : p ∈ P , λ ∈ F q ª ⊂ F d + q ,and Π ′ : = © s ( a , . . ., a d , − a d + ) : s ∈ F ∗ q , a x + · · · + a d x d = a d + ∈ Π ª . SPHERICAL EXTENSION THEOREM AND APPLICATIONS 9
Since P lies on the paraboloid P in F dq , one can check that P ′ lies on the variety definedby the equation x d + · x d = x + · · · + x d − ,which is the cone C d + after a change of variables. Thus, the argument in the case 1 willgive us the desired exponent.In other words, the case t = t >
1, one can partition the point set P into t subsets, each on a variety, then we canapply the case t = (cid:3)
4. A
N ENERGY BOUND
For a set A in F dq , the additive energy, denoted by E ( A ) is defined as the number of thepairs ( a , b , c , d ) ∈ A such that a + b = c + d . Theorem 4.1.
Suppose that d = k − and q ≡ . Let S j be a sphere of squareradius j in F dq . For A ⊂ S j , we haveE ( A ) ≪ | A | q + q d − | A | + q d − | A | . Proof.
We start with the following observation. Given a , b , c , d ∈ S j , if a + b = c + d , thenwe have ( b − d ) · ( a − d ) = d . Thus E ( A ) is bounded by the number of triples( a , b , d ) ∈ A such that ( b − d ) · ( a − d ) =
0. We now fall into two cases:
Case : Let E be the number of triples ( a , b , d ) ∈ A such that either || a − d || = || b − d || =
0. We are going to show that E ≪ | A | q + q d − | A | + q d − | A | .Indeed, if || a − d || =
0, then we have a · d = j . The identity a · d = j can be understood asan incidence between the point d ∈ A ⊂ S j and the hyperplane defined by the equation a · x = j . One can apply Corollary 2.4 to show that the number of pairs ( a , b ) ∈ A × A suchthat || a − d || = | A | q + O ( q d − | A | + q d − | A | ).By the same argument, this estimate holds in the case when || b − d || =
0. Thus, E ≪ | A | q + q d − | A | + q d − | A | . Case : Let E be the number of triples ( a , b , d ) ∈ A such that || a − d || 6= || b − d || 6= d ∈ A , we now count the number of pairs ( a , b ) ∈ A such that ( a − d ) · ( b − d ) = || a − d || 6=
0, there is no other point a ′ ∈ A such that a ′ − d = λ ( a − d ) for some λ ∈ F ∗ q \ { } . The same also holds for b − d .We observe that the identity ( a − d ) · ( b − d ) = a ∈ A and the hyperplane defined by ( b − d ) · x = ( b − d ) · d . Since A ⊂ S j , Corollary2.4 gives us that the number of pairs ( a , b ) ∈ A such that ( a − d ) · ( b − d ) = | A | q + O ( q d − | A | + q d − | A | ).Taking the sum over all d ∈ A , E ≪ | A | q + q d − | A | + q d − | A | .Putting E and E together, the theorem follows. ä (cid:3)
5. E
XTENSION THEOREMS FOR SPHERES
In this section, we give a complete proof of Theorem 1.4. We start this section with thefollowing lemma.
Lemma 5.1.
Suppose that d = k − and q ≡ . Let S j be a sphere of square radiusj in F dq . For A ⊂ S j of size n, we have k ( Ad σ ) ∨ k L ( F dq , dc ) ≪ n q − d + for q d + ≤ n ≪ q d − n q − d + for q d − ≤ n ≤ q d + n q − d + for q d − ≤ n ≤ q d − n q − d + for ≤ n ≤ q d − . Proof.
Using the orthogonal property of χ , we have k ( Ad σ ) ∨ k L ( F dq , dc ) = q d | S j | · E ( A ) ∼ q − d + E ( A ) .We now fall into two cases: Case : If q d − ≤ n ≪ q d − , then we can apply Theorem 4.1 to get the desired bounds. Case : If n ≤ q d − , then we use the trivial bound n for the energy to conclude theproof. (cid:3) SPHERICAL EXTENSION THEOREM AND APPLICATIONS 11
Proof of Theorem 1.4.
By a direct computation, Theorem 1.4 can be rephrased as fol-lows: k ( f d σ ) ∨ k L ( F dq , dc ) ≪ k f k L d /(3 d − ( S j , d σ ) ∼ Ã q − d + X x ∈ S j | f ( x ) | d d − ! d − d .Thus it suffices to prove the following inequality: q d − d + d k ( f d σ ) ∨ k L ( F dq , dc ) ≪ Ã X x ∈ S j | f ( x ) | d d − ! d − d .Without loss of generality, we may assume that the test function f is a nonnegative realvaled function since a general complex valued function f is written as the form f + i f for some real valued functions f and f , and a real valued function f can be expressedas the difference of two nonnegative real valued functions. Furtherore, normalizing thefunction f if necessary, we may assume that(5.1) X x ∈ S j | f ( x ) | d d − = T : = q d − d + d k ( f d σ ) ∨ k L ( F dq , dc ) ≪ f , k ( f d σ ) ∨ k L ( F dq , dc ) = q d | S j | X a , b , c , d ∈ A : a + b = c + d f ( a ) f ( b ) f ( c ) f ( d ).Hence, without loss of generality, we may assume that the test function f takes thefollowing form:(5.2) f ( x ) = ∞ X i = − i A i ( x ),where { A i } are disjoint subsets of S j . It follows from (5.1) and (5.2) that ∞ X i = − d d − i | A i | = | A i | ≤ d d − i , ∀ i .Let N = C log q , a positive number for some sufficiently large constant C . It follows that T ≤ q d − d + d N X i = − i k ( A i d σ ) ∨ k L ( F dq , dc ) + q d − d + d ∞ X i = N + − i k ( A i d σ ) ∨ k L ( F dq , dc ) = : M + R . We first bound R . Since | ( A i d σ ) ∨ ( m ) | ≤ m ∈ F dq , it is clear that k ( A i d σ ) ∨ k L ( F dq , dc ) ≤ q d /4 . It therefore follows that R ≤ q d − d + d q d ∞ X i = N + − i ≪ q d − d + d − N ≪ M . To do this, we decompose the sum P Ni = as four subsummands asfollows: N X i = = X ≤ i ≤ N ≤ d d − i ≤ q d − + X ≤ i ≤ Nq d − ≤ d d − i ≤ q d − + X ≤ i ≤ Nq d − ≤ d d − i ≤ q d + + X ≤ i ≤ Nq d + ≤ d d − i ≪ q d − = : X + X + X + X .Then, with notations above, the term M is written by M = q d − d + d X − i k ( A i d σ ) ∨ k L ( F dq , dc ) + q d − d + d X − i k ( A i d σ ) ∨ k L ( F dq , dc ) + q d − d + d X − i k ( A i d σ ) ∨ k L ( F dq , dc ) + q d − d + d X − i k ( A i d σ ) ∨ k L ( F dq , dc ) = : M + M + M + M .Employing Lemma 5.1 and using (5.3), we get M ≪ q − d + d X ≤ i ≤ N ≤ d d − i ≤ q d − − i | A i | ≤ q − d + d X ≤ i ≤ N ≤ d d − i ≤ q d − d − i ≪ q − d + d · q d − d = M ≪ q d − d + d X ≤ i ≤ Nq d − ≤ d d − i ≤ q d − − i | A i | ≤ q d − d + d X ≤ i ≤ Nq d − ≤ d d − i ≤ q d − − d + d − i ≪ q d − d + d · q − d + d − d = M ≪ q − d + d X ≤ i ≤ Nq d + ≤ d d − i ≪ q d − − i | A i | ≤ q − d + d X ≤ i ≤ Nq d + ≤ d d − i ≪ q d − d − i ≪ q − d + d · q d − d = M ≪
1, which will be proven separately in the cases of d = d = k − ≥
7. As before, it follows by Lemma 5.1 and (5.3) that M ≪ q d − d + d X ≤ i ≤ Nq d − ≤ d d − i ≤ q d + − i | A i | ≤ q d − d + d X ≤ i ≤ Nq d − ≤ d d − i ≤ q d + − d + d − i . SPHERICAL EXTENSION THEOREM AND APPLICATIONS 13
Hence, if d =
3, then M ≪ q − X ≤ i ≤ Nq ≤ i ≤ q i ≪ q − q = d ≥
7, then we have M ≪ q d − d + d X ≤ i ≤ Nq d − ≤ d d − i ≤ q d + − d + d − i ≪ q d − d + d q − d + d − d = q − d + d ≤ ä
6. T
HREE - DISTANCE PROBLEM (T HEOREM f on F dq is defnied by b f ( x ) = X m ∈ F dq χ ( − m · x ) f ( m ),where χ denotes a nontrivial additive character of F q . This definition should be comparedwith the definition of the Fourier transform b f used in other papers. We emphasize thatthere does not appear a normalizing factor q − d in the definition of b f , while such a nor-malizing factor has been used in many other articles such as [12], [7], [1], and [2]. Recallthat the Fourier inversion theorem states that f ( m ) = q − d X x ∈ F dq χ ( m · x ) b f ( x ).To prove Theorem 1.6, we need the following results. The first proposition is known asthe interpolation proposition. A detailed proof can be found in [5]. Proposition 6.1.
Let ≤ r , r , p , p ≤ ∞ with r ≤ r and p ≤ p . (1) Suppose that T is an linear operator and the following two estimates hold for allfunctions f : k T f k L r ≤ C and k T f k L r ≤ C . Then we have k T f k L r ≤ C − θ C θ for any ≤ θ ≤ with − θ r + θ r = r .(2) Suppose that T is an linear operator and the following two estimates hold for allfunctions f : || T f || L r ≤ C || f || L p , || T f || L r ≤ C || f || L p . Then we have || T f || L r ≤ C − θ C θ || f || L p , for any ≤ θ ≤ with p = − θ p + θ p , 1 r = − θ r + θ r .For t ∈ F q , let µ ( t ) be the number of triples ( x , y , z ) ∈ A such that || x + y + z || = t . We recallthe following lemma from [3]. Lemma 6.2 ([3], Lemma 2.6, Lemma 2.7) . Let A ⊂ F dq with d ≥ even. If | A | ≥ q d /2 , thenwe have ¡ | A | − µ (0) ¢ ≥ | A | and q − d ¯¯¯¯¯ X x ∈ S ¡ b A ( x ) ¢ ¯¯¯¯¯ − µ (0) ≤ | A | q . Here, we note that b A ( x ) = P m ∈ A χ ( − x · m ).We have the following lemma on the average of the second moment of function µ ( t ). Lemma 6.3.
Let A be a set in F dq with d even. (1) Suppose that | A | ≥ q d /2 and d = , then we have X t ∈ F ∗ q µ ( t ) ≪ | A | q + q | A | .(2) Suppose that | A | ≥ q d /2 and d ≥ , then we have X t ∈ F ∗ q µ ( t ) ≪ | A | q + q d − d + d − | A | − d d − .Now we will make a reduction for the proof of Lemma 6.3. We observe that µ ( t ) = X x , y , z ∈ F dq A ( x ) A ( y ) A ( z ) S t ( x + y + z ) = q d · X x , y , z ∈ F dq A ( x ) A ( y ) A ( z ) X m ∈ F dq c S t ( m ) χ ( m · ( x + y + z )) = q d · X m ∈ F dq c S t ( m ) ³ b A ( m ) ´ .Thus, we have X t ∈ F q µ ( t ) = X t ∈ F q µ ( t ) µ ( t ) = q d · X m , v ∈ F dq X t ∈ F q c S t ( m ) c S t ( v ) ³ b A ( m ) ´ ¡ b A ( v ) ¢ .(6.1) SPHERICAL EXTENSION THEOREM AND APPLICATIONS 15
We also make use of the following lemma which is taken from [15, Proposition 2.2].
Lemma 6.4.
For m , v ∈ F dq , we have X t ∈ F q c S t ( m ) c S t ( v ) = q d · ³ δ ( m ) δ ( v ) q + q − ( d + X s χ ( s · ( || m || − || v || )) ´ .Substituting this bound to (6.1), we get X t ∈ F q µ ( t ) = q ¯¯ b A (0) ¯¯ + q d + · X m , v ∈ F dq ³ b A ( m ) ´ ¡ b A ( v ) ¢ à X s ∈ F q χ ( s ( || m || − || v || )) − ! = | A | q + q d · X || m ||=|| v || ³ b A ( m ) ´ ¡ b A ( v ) ¢ − q d + ¯¯¯¯¯¯ X v ∈ F dq ¡ b A ( v ) ¢ ¯¯¯¯¯¯ ≤ | A | q + q d · X || m ||=|| v || ³ b A ( m ) ´ ¡ b A ( v ) ¢ ≤ | A | q + q d · X r ∈ F q ¯¯¯¯¯ X || v ||= r ¡ b A ( v ) ¢ ¯¯¯¯¯ Lemma 6.2 tells us that X t µ ( t ) ≤ | A | q + q d · à max r X v ∈ S r ¯¯ b A ( v ) ¯¯ ! X v ∈ F dq ¯¯ b A ( v ) ¯¯ ≤ | A | q + | A | à max r X v ∈ S r ¯¯ b A ( v ) ¯¯ ! ,where we have used the Hölder inequality and the facts that P v ∈ F dq | b A ( v ) | = q d | A | , | b A ( v ) | ≤| b A (0) | = | A | to bound the sum P v ∈ F dq ¯¯ b A ( v ) ¯¯ .Therefore, in order to prove Lemma 6.3, it is enough to address the following theorem,which will be shown using L p → L spherical restriction estimates. Theorem 6.5.
For A ⊂ F dq with d even, the following statements hold. (1) Suppose that d = , then we have || b A || L ( S t , d σ ) ≪ | A | for any t Suppose that d ≥ , then we have || b A || L ( S t , d σ ) ≪ q − d − d + d − | A | − d d − for any t Proof.
Case : For any function f : S t → C , we recall the trivial bound || ( f d σ ) ∨ || L ∞ ( F q , dc ) ≪ || f || L ( S t , d σ ) .It was also proved in [11, Theorem 1.5] that || ( f d σ ) ∨ || L ( F q , dc ) ≪ || f || L ( S t , d σ ) .Using Proposition 6.1, we have || ( f d σ ) ∨ || L ( F q , dc ) ≪ || f || L ( S t , d σ ) .By duality, one has || b g || L ( S t , d σ ) ≪ || g || L ( F q , dc ) ,for all function g : F q → C . Set g being the characteristic function of A , then the statementfollows. Case : To prove this case, we first need to show that || b A || L ( S t , d σ ) ≪ | A | q d − ,whenever | A | ≥ q d − . Indeed, || b A || L ( S t , d σ ) = | S t | X x ∈ S t | b A ( x ) | ∼ q d − X x ∈ S t | b A ( x ) | .Thus, it is enough to handle the following inequality X x ∈ S t | b A ( x ) | ≪ q d − | A | .It follows from the definition of b A ( x ) that X x ∈ S t | b A ( x ) | = X x ∈ S t X a , b ∈ A χ ( − x ( a − b )) = X a , b ∈ A c S t ( a − b ) = | A | c S t (0) + X a , b ∈ A , a b c S t ( a − b ) ≤ | A || S t | + X a , b ∈ A , a b µ max x | c S t ( x ) | ¶ .Moreover, it has been indicated in [12] thatmax x | c S t ( x ) | = max x ¯¯¯¯¯ X m ∈ S t χ ( − x · m ) ¯¯¯¯¯ ≪ q d − .Thus, we obtain X x ∈ S t | b A ( x ) | ≪ q d − | A | + q d − | A | ≪ q d − | A | ,under the condition | A | ≥ q d − . SPHERICAL EXTENSION THEOREM AND APPLICATIONS 17
Since d is even, by duality, it was proved in [11, Theorem 1.5] that || b A || L dd + ( S t , d σ ) ≪ k A k L ( F dq , dc ) = | A | .Thus, if d ≥
2, and | A | ≥ q d − , using the Proposition 6.1 with θ = d /(3 d − || b A || L ( S t , d σ ) ≪ ³ q − d − | A | ´ − θ | A | θ .Notice that since d ≥
6, we have 0 ≤ θ ≤
1. Hence, || b A || L ( S t , d σ ) ≪ q − d − d + d − | A | − d d − .This completes the proof of the theorem. (cid:3) Proof of Theorem 1.6: Using the Cauchy-Schwarz inequality and the first statement ofLemma 6.2, we have | ∆ ( A ) | ≫ | A | P t µ ( t ) .On the other hand, Lemma 6.3 gives us X t µ ( t ) ≪ | A | q + q | A | , d = | A | q + q d − d + d − | A | − d d − , d ≥ | ∆ ( A ) | ≫ q ,whenever | A | ≫ ( q , d = q d + d − d − , d ≥ ä A CKNOWLEDGEMENTS
D. Koh was supported by Basic Science Research Program through the National Re-search Foundation of Korea(NRF) funded by the Ministry of Education, Science andTechnology(NRF-2018R1D1A1B07044469). T. Pham was supported by Swiss NationalScience Foundation grant P400P2–183916.The authors would like to thank the Vietnam Institute for Advanced Study in Mathemat-ics for hospitality during their visit. R
EFERENCES
1. J. Chapman, M. Burak Erdo ˘gan, D. Hart, A. Iosevich, and D. Koh,
Pinned distance sets, k-simplices,Wolff’s exponent in finite fields and sum-product estimates , Math Z. (2012), no. 1, 63-93.
2. D. Covert, D. Koh, and Y. Pi,
The generalized k-resultant modulus set problem in finite fields , J. FourierAnal. Appl. (2019), no. 3, 1026-1052.3. D. Covert, D. Koh, and Y. Pi, On the sums of any k points in finite fields , SIAM J. Discrete Math, (2016), no.1, 367-382.4. K. J. Falconer, On the Hausdorff dimensions of distance sets , Mathematika, (1985), no. 2, 206–212(1986).5. L. Grafakos, Classical and modern Fourier analysis , Pearson. Education, Inc. (2004).6. W. H. Haemers,
Eigenvalue techniques in design and graph theory , Number 121, Mathematisch cen-trum Amsterdam, 1980.7. D. Hart, A. Iosevich, D. Koh, and M. Rudnev,
Averages over hyperplanes, sum-product theory in vectorspaces over finite fields and the Erd˝os-Falconer distance conjecture , Trans. Amer. Math. Soc. (2011),no. 6, 3255-3275.8. A. Iosevich and D. Koh,
Extension theorems for the Fourier transform associated with non-degeneratequadratic surfaces in vector spaces over finite fields , Illinois J. of Mathematics, (2008), no.2, 611–628.9. A. Iosevich and D. Koh, Extension theorems for spheres in the finite field setting , Forum. Math. (2010), no.3, 457–483.10. A. Iosevich, D. Koh, and M. Lewko, Finite field restriction estimates for the paraboloid in high evendimensions,
J. Funct. Anal. (2020), no.11, 108450.11. A. Iosevich, D. Koh, S. Lee, T. Pham, and C-Y. Shen,
On restriction estimates for the zero radius sphereover finite fields , Canad. J. Math. to appear (2020).12. A. Iosevich, M. Rudnev,
Erd˝os–Falconer distance problem in vector spaces over finite fields , Trans. Amer.Math. Soc. (2007), no.12, 6127-6142.13. D. Koh, S. Lee, and T. Pham,
On the cone restriction conjecture in four dimensions and applications inincidence geometry , arXiv:2004.06593 (2020).14. D. Koh, T. Pham, and L. A. Vinh,
Extension theorems and a connection to the Erd˝os-Falconer distanceproblem over finite fields , arXiv:1809.08699, (2018).15. D. Koh and H. Sun,
Distance sets of two subsets of vector spaces over finite fields , Proc. Amer. Math.Soc. (2015), no. 4, 1679-1692.16. M. Lewko,
New restriction estimates for the 3-d paraboloid over finite fields , Adv. Math. (2015), no.1, 457-479.17. M. Lewko,
Finite field restriction estimates based on Kakeya maximal operator estimates,
J. Eur. Math.Soc. (2019), no.12, 3649-3707.18. M. Lewko, Counting rectangles and an improved restriction estimate for the paraboloid in F p , Proc.Amer. Math. Soc. (2020), no.4, 1535-1543.19. R. Lidl and H. Niederreiter, Finite fields , Cambridge University Press, (1997).20. B. Murphy, G. Petridis, T. Pham, M. Rudnev, and S. Stevens,
On the Pinned Distances Problem overFinite Fields , arXiv:2003.00510 (2020).21. G. Mockenhaupt and T. Tao,
Restriction and Kakeya phenomena for finite fields , Duke Math. J. (2004), no.1, 35-74.22. M. Rudnev and I. Shkredov,
On the restriction problem for discrete paraboloid in lower dimension,
Adv.Math. (2018), 657-671.23. Le Anh Vinh,
The Szemerédiâ ˘A ¸STrotter type theorem and the sum-product estimate in finite fields ,European J. Combin. (2011) no.8, 1177â ˘A ¸S1181. SPHERICAL EXTENSION THEOREM AND APPLICATIONS 19 D EPARTMENT OF M ATHEMATICS , C
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