Sums and products in finite fields: an integral geometric viewpoint
aa r X i v : . [ m a t h . N T ] J un Sums and products in finite fields: an integral geometricviewpoint
Derrick Hart and Alex IosevichOctober 25, 2018
Abstract
We prove that if A ⊂ F q is such that | A | > q + d , then F ∗ q ⊂ dA = A + · · · + A d times , where A = { a · a ′ : a, a ′ ∈ A } , and where F ∗ q denotes the multiplicative group of the finite field F q . In particular, wecover F ∗ q by A + A if | A | > q . Furthermore, we prove that if | A | ≥ C d size q + d − , then | dA | ≥ q · C size C size + 1 . Thus dA contains a positive proportion of the elements of F q under a considerablyweaker size assumption.We use the geometry of F dq , averages over hyper-planes andorthogonality properties of character sums. In particular, we see that using opera-tors that are smoothing on L in the Euclidean setting leads to non-trivial arithmeticconsequences in the context of finite fields. Contents
Proof of the basic geometric estimate (Theorem 1.4) 53 Proof of the enhanced geometric estimate (Theorem 1.5) 74 Proof of the main arithmetic result (Theorem 1.1) 9
A classical problem in additive number theory is to determine, given a finite subset A of aring, whether both 2 A = { a + a ′ : a, a ′ ∈ A } and A = { a · a ′ : a, a ′ ∈ A } can be small in asuitable sense. A related question, posed in a finite field F q with q elements, is how large A ⊂ F q need to be to assure that dA = A + A + · · · + A = F q . It is known (see e.g.[6]) that if d = 3 and q is prime, this conclusion is assured if | A | ≥ Cq , with a sufficientlylarge constant C >
0. It is reasonable to conjecture that if | A | ≥ C ǫ q + ǫ , then 2 A = F q .This result cannot hold, especially in the setting of general finite fields if | A | = √ q because A may in fact be a subfield. See also [1], [3], [5], [4], [8], [10], [12], [13] and the referencescontained therein on recent progress related to this problem and its analogs.For example, it is proved in [5] that8 X · Y = Z p , for p prime, provided that | X || Y | > p and either Y = − Y or Y ∩ ( − Y ) = ∅ . In [6] theauthor prove that if A is subgroup of Z ∗ p , and | A | > p δ , δ >
0, then
N A = Z p with N ≤ C δ . The purpose of this paper is to use the geometry of F dq , where q is not necessarily a primenumber, to deduce a good lower bound on the size of A that guarantees that dA = F q ,with the possible exception of 0. Furthermore, it is shown that the lower bound on A maybe relaxed if one settles for a positive proportion of F q . Our main result is the following. Theorem 1.1.
Let A ⊂ F q , where F q is an arbitrary finite field with q elements, such that | A | > q + d . Then F ∗ q ⊂ dA . (1.1) Suppose that | A | ≥ C d size q + d − . Then | dA | ≥ q · C − d size C − d size + 1 . (1.2)2n particular, if d = 2, F ∗ q ⊂ A + A if | A | > q , and | A + A | ≥ q · C size C size + 1if | A | ≥ C size q . Also, Theorem 1.1 gives an explicit bound for the conjecture mentioned in [6], namelythat if | A | ≥ C ǫ q + ǫ , there exists d = d ( ǫ ) such that dA covers F q . In view of this, werestate Theorem 1.1 as follows. Theorem 1.2.
Let A ⊂ F q , where F q is an arbitrary finite field with q elements, such that | A | ≥ C ǫ q + ǫ , for some ǫ > . Then (1.1) holds for d = d ( ǫ ) equal to the smallest integer greater than orequal to ǫ . Moreover, (1.2) holds if d is equal to the smallest integer greater than or equalto + ǫ . Throughout the paper, X . Y means that there exists a universal constant C , indepe-dent of q , such that X ≤ CY , and X ≈ Y means that X . Y and Y . X . In the instanceswhen the size of the constant matters, this fact shall be mentioned explicitly. Remark . The reader can easily check that in Theorem 1.1 and Theorem 1.2, dA maybe easily replaced by A · B + · · · + A d · B d , provided that Π dj =1 | A j || B j | ≥ Cq d +1 with a sufficiently large constant C >
Theorem 1.4.
Let E ⊂ F dq such that | E | > q d +12 . Then F ∗ q ⊂ { x · y : x, y ∈ E } . To prove Theorem 1.1 we shall need the following conditional version of Theorem 1.4.3 heorem 1.5.
Let E ⊂ F dq such that | E ∩ l y | ≤ C geom q αd for some ≤ α ≤ d , for every y ∈ F dq , y = (0 , . . . , , where l y = { ty : t ∈ F q } . Suppose that | E | ≥ C size q d + α d . Then |{ x · y : x, y ∈ E }| ≥ q · C size C size + C geom . Remark . Theorem 1.5 has non-trivial applications to many other problems in additivenumber theory and geometric combinatorics, such as the Erd˝os distance problem, distri-bution of simplexes and others. We study these problems systematically in [7].
At the core of the proof of Theorem 1.4 and Theorem 1.5 is the L ( F dq ) estimate for the”rotating planes” operator R t f ( x ) = X x · y = t f ( y ) . In the Euclidean space, this operator is a classical example of a phenomenon, thor-oughly explored by Hormander, Phong, Stein and others (see e.g. [11]) and the referencescontained therein) where an operator that averages a function over a family of manifoldssatisfies better than trivial bounds on L ( F dq ) provided that the family of manifolds sat-isfies an appropriate curvature condition. It turns out that in the finite field setting, theaforementioned operator, suitably interpreted, satisfies analogous bounds which lead tointeresting arithmetic consequences.In contrast, the authors of [8] took advantage of the L ( F dq ) mapping properties of theoperator H j f ( x ) = X y y = j f ( x − y ) , and in [9] the underlying operator is A t f ( x ) = X y + ··· + y d = t f ( x − y ) , though in neither paper was this perspective made explicit. These examples suggest thatsystematic theory of Fourier Integral Operator in the setting of vector spaces over finitefields needs to be worked out and the authors shall take up this task in a subsequent paper.4 .2 Fourier analysis used in this paper Let f : F dq → C . Let χ be a non-trivial additive character on F q . Define the Fouriertransform of f by the formula b f ( m ) = q − d X x ∈ F dq χ ( − x · m ) f ( x )for m ∈ F dq .The formulas we shall need are the following: X t ∈ F q χ ( − at ) = 0 (orthogonality) , if t = 0, and q otherwise, f ( x ) = X m χ ( x · m ) b f ( m ) (inversion) , X m b f ( m ) b g ( m ) = q − d X x f ( x ) g ( x ) (Plancherel/Parseval) . In the case when q is a prime, one may take χ ( t ) = e πiq t , and in the general case theformula is only slightly more complicated. The authors wish to thank Moubariz Garaev, Nets Katz, Sergei Konyagin and IgnacioUriarte-Tuero for a thorough proofreading of the earlier drafts of this paper and for manyinteresting and helpful remarks.
Let ν ( t ) = |{ ( x, y ) ∈ E × E : x · y = t }| . We have ν ( t ) = X x,y ∈ E q − X s ∈ F q χ ( s ( x · y − t )) , where χ is a non-trivial additive character on F q . It follows that ν ( t ) = | E | q − + R, R = X x,y ∈ E q − X s =0 χ ( s ( x · y − t )) . Viewing R as a sum in x , applying the Cauchy-Schwartz inequality and dominating thesum over x ∈ E by the sum over x ∈ F dq , we see that R ≤ | E | X x ∈ F dq q − X s,s ′ =0 X y,y ′ ∈ E χ ( sx · y − s ′ x · y ′ ) χ ( t ( s ′ − s )) . Orthogonality in the x variable yields= | E | q d − X sy = s ′ y ′ s,s ′ =0 χ ( t ( s ′ − s )) E ( y ) E ( y ′ ) . If s = s ′ we may set a = s/s ′ , b = s ′ and obtain | E | q d − X y = y ′ ay = y ′ a =1 ,b χ ( tb (1 − a )) E ( y ) E ( y ′ )= −| E | q d − X y = y ′ ,a =1 E ( y ) E ( ay ) , and the absolute value of this quantity is ≤ | E | q d − X y ∈ E | E ∩ l y |≤ | E | q d − , since | E ∩ l y | ≤ q by the virtue of the fact that each line contains exactly q points.If s = s ′ we get | E | q d − X s,y E ( y ) = | E | q d − . It follows that ν ( t ) = | E | q − + R ( t ) , where R ( t ) ≤ − Q ( t ) + | E | q d − , Q ( t ) ≥ . It follows that R ( t ) ≤ | E | q d − , so | R ( t ) | ≤ | E | q d − . (2.1)We conclude that ν ( t ) = | E | q − + R ( t )with | R ( t ) | bounded as in (2.1).This quantity is strictly positive if | E | > q d +12 with a sufficiently large constant C > E = A × A × · · · × A . Assume throughout the argument, without loss of generality, that E does not contain theorigin. Applying Cauchy-Schwartz as above we see that ν ( t ) ≤ | E | X x ∈ E X y,y ′ ∈ E q − X s,s ′ χ ( x · ( sy − s ′ y ′ )) χ ( t ( s ′ − s )) . It follows that X t ν ( t ) ≤ | E | q d − X s X m b E ( sm ) X y − y ′ = m E ( y ) E ( y ′ )= | E | q d − X s X m b E ( ms ) E ∗ E ( m )= | E | q d − X m X s b E ( sm ) ! E ∗ E ( m ) . (3.1)Now, X s b E ( ms ) = X s q − d X x E ( x ) χ ( − x · ms )= q − ( d − X x · m =0 E ( x ) . | E | X m X x · m =0 E ( x ) ! · E ∗ E ( m ) . (3.2)Let F ( m ) = X x · m =0 E ( x ) , G ( m ) = E ∗ E ( m ) . By a direct calculation, b G ( k ) = q d | b E ( k ) | . On the other hand, b F ( k ) = q − d X m χ ( − m · k ) X x · m =0 E ( x )= q − d q − X m,x X s χ ( − m · k + sx · m ) E ( x )= q − d q − X m,x X s =0 χ ( − m · k + sx · m ) E ( x )= q − X s =0 E ( s − k )= q − X s =0 E ( sk ) = q − | E ∩ l k | , if k = (0 , . . . ,
0) and q − | E | , if k = (0 , . . . , | E | X m F ( m ) G ( m ) = | E | q d X k b F ( k ) b G ( k )= | E | q d − X k =(0 ,..., | E ∩ l k || b E ( k ) | + | E | q d − · | E | · q − d | E | ≤ C geom | E | q d − q αd q − d | E | + | E | q − C geom | E | q d − αd + | E | q − . Since | E | = X t ν ( t ) ! ≤ |{ x · y : x, y ∈ E }| · X t ν ( t )8 |{ x · y : x, y ∈ E }| (cid:16) C geom | E | q d − αd + | E | q − (cid:17) , it follows that |{ x · y : x, y ∈ E }| ≥ q · | E | C geom q d + αd + | E | = r q · q. (3.3)Suppose that | E | ≥ C size q d + α d . It follows that r q ≥ C size C size + C geom , as desired. Let E = A × A × · · · × A . The proof of the first part of Theorem 1.1 follows instantly. Toprove the second part observe that | E ∩ l y | ≤ | A | = | E | d for every y ∈ E .Then the line (3.3) takes the form |{ ( x · y : x, y ∈ E }| ≥ q · | E | q d · | E | d + | E | . The proof of Theorem 1.5 tells us at this point that |{ x · y : x, y ∈ E }| ≥ q · C − d size C − d size + 1if | E | ≥ C size q d + d d − . It follows that if | A | ≥ C d size q + d − , then | dA | ≥ q · C − d size C − d size + 1as desired. This completes the proof of Theorem 1.1.9 eferences [1] J. Bourgain, A. A. Glibichuk and S. V. Konyagin, Estimates for the number of sumsand products and for exponential sums in fields of prime order , J. London Math. Soc.(2) (2006), 380-398.[2] J. Bourgain, N. Katz and T. Tao, A sum-product estimate in finite fields, and appli-cations , Geom. Func. Anal. (2004) 27-57.[3] E. Croot, Sums of the Form /x k + . . . /x kn modulo a prime , Integers (2004).[4] M. Garaev, The sum-product estimate for large subsets of prime fields , (preprint),(2007).[5] A. A. Glibichuk,
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