Points on polynomial curves in small boxes modulo an integer
aa r X i v : . [ m a t h . N T ] M a r POINTS ON POLYNOMIAL CURVES IN SMALLBOXES MODULO AN INTEGER
BRYCE KERR AND ALI MOHAMMADI
Abstract.
Given an integer q and a polynomial f ∈ Z q [ X ] ofdegree d with coefficients in the residue ring Z q = Z /q Z , we obtainnew results concerning the number of solutions to congruences ofthe form y ≡ f ( x ) (mod q ) , with integer variables lying in some cube B of side length H . Ourargument uses ideas of Cilleruelo, Garaev, Ostafe and Shparlinskiwhich reduces the problem to the Vinogradov mean value theoremand a lattice point counting problem. We treat the lattice pointproblem differently using transference principles from the Geom-etry of Numbers. We also use a variant of the main conjecturefor the Vinogradov mean value theorem of Bourgain, Demeter andGuth and of Wooley which allows one to deal with rather sparsesets. Introduction
Given an integer q and a polynomial f ∈ Z q [ X ] of degree d withcoefficients in the residue ring Z q = Z /q Z , we consider the problem ofestimating the number of solutions to congruences of the form y ≡ f ( x ) (mod q ) , (1)with integer variables( x, y ) ∈ ( K, K + H ] × ( L, L + H ] , lying in some cube of side length H . This problem and its variants havebeen considered by a number of previous authors, see for example [8,9, 10, 11, 12, 13, 14, 18, 19]. A common strategy for dealing with suchequations is to lift (1) to a polynomial equation over Z to which onemay apply the following result of Bombieri and Pila [2]. Theorem 1.
Let C be an absolutely irreducible curve of degree d > and suppose H > e d . The number of integral points on C and inside a Date : March 29, 2018. square [1 , H ] × [1 , H ] is bounded by O (cid:0) H /d exp (12(log H log log H ) / ) (cid:1) . To see how one may lift the equation (1) to an equation over Z , wefirst note that by shifting variables and modifying the coefficients of f we may assume K = L = 0. Now, suppose f is given by f ( x ) = a + a x + · · · + a d x d , with leading coefficient satisfying ( a d , q ) = 1. We define the lattice L = { ( z, w , . . . , w d ) ∈ Z d +1 : ∃ n ∈ Z such that n ≡ z (mod q ) , a i n ≡ w i (mod q ) } , and the convex body D = n ( z, w , . . . , w d ) ∈ Z d +1 : | z | qH , | w i | qH i o . By Minkowski’s first theorem, D contains a non-zero lattice point of L provided(2) Vol( D ) > d +1 Vol (cid:0) R d +1 / L (cid:1) . Since Vol( R d +1 / L ) = q d , and Vol( D ) = 2 d +1 q d +1 H ( d + d +2) / , we see that (2) is satisfied provided H q / ( d + d +2) . (3)Suppose H satisfies (3), let ( z, w , . . . , w d ) be a nonzero lattice pointof L ∩ D and suppose n is defined by n ≡ z (mod q ) , a i n ≡ w i (mod q ) . If w denotes the smallest residue of a n (mod q ) then the congruence y ≡ f ( x ) (mod q ) , is equivalent to zy ≡ w + w x + · · · + w d x d (mod q ) , and hence each point satisfying (1) must also satisfy zy = w + w x + · · · + w d x d + tq, OINTS ON POLYNOMIAL CURVES IN SMALL BOXES 3 for some t ∈ Z . Since ( a d , q ) = 1 we have w d = 0. From ( z, w , . . . , w d ) ∈ D we see that there are O (1) possible values for t when the variables x, y satisfy 1 x H and 1 y H, and for each such value of t we apply Theorem 1. This results in thefollowing. Theorem 2.
Let q be an integer, f ∈ Z q [ X ] a polynomial of degree d > with leading coefficient coprime to q and suppose B is a cube ofside length H . If H q / ( d + d +2) , then |{ ( x, y ) ∈ B : y ≡ f ( x ) (mod q ) }| H /d + o (1) . In this paper we consider the problem of determining the largestnumber α depending only on d such that for all integers q and poly-nomials f ∈ Z q [ X ] of degree d with leading coefficient coprime to q wehave the following estimate |{ ( x, y ) ∈ B : y ≡ f ( x ) (mod q ) }| H /d + o (1) , (4)uniformly over all cubes B of side length H q α . When q is prime, im-provements on Theorem 2 are known. For example, Cilleruelo, Garaev,Ostafe and Shparlinski [13, Theorem 3] have shown that (4) holds pro-vided H q / ( d +3) . We obtain new results concerning estimates for the number of solu-tions to equations of the form (1). In particular, we improve on theabove result of Cilleruelo, Garaev, Ostafe and Shparlinski [13, Theo-rem 3] and some results of Chang, Cilleruelo, Garaev, Hernndez, Sh-parlinski and Zumalac´arregui [10, Corollary 3]. Our argument usesideas from [10] and [13] which reduces the problem of bounding thenumber of solutions to (1) to the Vinogradov mean value theorem anda problem concerning the number of lattice points in the intersectionof a convex body. We treat the lattice point problem differently us-ing transference principles from the Geometry of Numbers. We alsouse some recent progress concerning the main conjecture for the Vino-gradov mean value theorem first obtained by Bourgain, Demeter andGuth [4] and shortly after by Wooley [17] which allows one to dealwith solutions to the Vinogradov mean value theorem when the vari-ables belong to rather sparse sets, see Lemma 8 below.
B. KERR AND A. MOHAMMADI
Finally, we mention techniques from the geometry of numbers havefound applications to a wide range of other problems concerning equa-tions in finite fields and residue rings to which we refer the readerto [3, 5, 6, 7, 15]. 2.
Main results
Our first result improves on a bound of Cilleruelo, Garaev, Ostafeand Shparlinski [13, Theorem 3].
Theorem 3.
Let q be an arbitrary integer, f ∈ Z q [ X ] a polynomialof degree d > with leading coefficient a d satisfying ( a d , q ) = 1 andsuppose B is a cube of side length H . We have |{ ( x, y ) ∈ B : y ≡ f ( x ) (mod q ) }| H /d ( d +1)+ o (1) q /d ( d +1) + H /d + o (1) . In particular, if H q / ( d +1) , then |{ ( x, y ) ∈ B : y ≡ f ( x ) (mod q ) }| H /d + o (1) . (5)For comparison with Theorem 3 we note that [13, Theorem 3] givesthe bound (5) in the shorter range H q / ( d +3) . Our second result improves on some results of [10].
Theorem 4.
Let q be an arbitrary integer, f ∈ Z q [ X ] of degree d = 3 and leading coefficient a satisfying ( a , q ) = 1 and B a cube of sidelength H . Then we have |{ ( x, y ) ∈ B : y ≡ f ( x ) (mod q ) }| H / q / + H / o (1) . In particular if H q / then |{ ( x, y ) ∈ B : y ≡ f ( x ) (mod q ) }| H / o (1) . For comparison with Theorem 4, we note that, for a prime p , thebounds of [10, Theorem 1] and [10, Theorem 2] imply that |{ ( x, y ) ∈ B : y ≡ f ( x ) (mod p ) }| H / o (1) , if H < p / , (cid:16) H p (cid:17) / H o (1) , if p / H < p / , (cid:16) H p (cid:17) / H o (1) , if p / H < p / . (6) OINTS ON POLYNOMIAL CURVES IN SMALL BOXES 5
We remark that one may incorporate Lemma 9, with ( k, s ) = (3 , |{ ( x, y ) ∈ B : y ≡ f ( x ) (mod p ) }| H / o (1) + (cid:18) H p (cid:19) / H o (1) . This improves on the bound (6) in the range p / H < p / andappears to be the best bound possible through the arguments of [10,Theorem 2]. We see that Theorem 4 improves on the bounds (6) and(7) in the range p / H < p / . Acknowledgement
The authors would like to thank Igor Shparlinski for his commentsand for pointing out an error in a previous version of the current paper.3.
Preliminary results
We first recall some facts from the geometry of numbers. Given alattice Γ ⊂ R n and a symmetric convex body D ⊂ R n we define the i -th successive minimum of Γ with respect to D by λ i = inf { λ : Γ ∩ λD contains i linearly independent points } . We define the dual lattice Γ ∗ byΓ ∗ = { y ∈ R n : h y, z i ∈ Z for all z ∈ Γ } , and the dual body D ∗ by D ∗ = { y ∈ R n : h y, z i z ∈ D } , where h ., . i denotes the Euclidean inner product. The following isMinkowski’s second theorem, for a proof see [16, Theorem 3.30]. Lemma 5.
Let Γ ⊂ R n be a lattice, D ⊂ R n a symmetric convex bodyand let λ , . . . , λ n denote the successive minima of Γ with respect to D .We have Vol ( D ) Vol ( R n / Γ) ≪ λ . . . λ n ≪ Vol ( D ) Vol ( R n / Γ) . We may bound the number of lattice points | Γ ∩ D | in terms of thesuccessive minima, see for example [16, Exercise 3.5.6]. B. KERR AND A. MOHAMMADI
Lemma 6.
Let Γ ⊂ R n be a lattice, D ⊂ R n a symmetric convex bodyand let λ , . . . , λ n denote the successive minima of Γ with respect to D .We have | Γ ∩ D | ≪ n Y j =1 max (cid:18) , λ j (cid:19) . The successive minima of a lattice with respect to a convex bodyand the successive minima of the dual lattice with respect to the dualbody are related through the following estimates, see for example [1].
Lemma 7.
Let Γ ⊂ R n be a lattice, D ⊂ R n a symmetric convex bodyand let λ , . . . , λ n denote the successive minima of Γ with respect to D .Let Γ ∗ denote the dual lattice, D ∗ the dual body and let λ ∗ , . . . , λ ∗ n denotethe successive minima of Γ ∗ with respect to D ∗ . For each j n we have λ j λ ∗ n − j +1 ≪ . The following is a consequence of results of Bourgain, Demeter andGuth [4, Theorem 4.1] and of Wooley [17, Theorem 1.1].
Lemma 8.
Let k be an integer, s a positive real number with s k ( k + 1) / and ( a n ) n ∈ Z a sequence of complex numbers. We have Z [0 , k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | n | X a n e ( α n + · · · + α k n k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s d α ≪ X o (1) X | n | X | a n | s . The following is an immediate corollary of Lemma 8.
Lemma 9.
Let
X ⊂ [1 , X ] ∩ Z be some set. For integers k and s welet J k,s ( X ) denote the number of solutions to the system of equations x j + · · · + x js = x js +1 + · · · + x j s , j k, with variables satisfying x , . . . , x s ∈ X . For s k ( k + 1) / we have J k,s ( X ) ≪ |X | s X o (1) . Proof of Theorem 3
Supposing the cube B is given by B = ( K, K + H ] × ( L, L + H ] , OINTS ON POLYNOMIAL CURVES IN SMALL BOXES 7 we may assume
K, L = 0 by applying shifts x → x − K , y → y − L and modifying the coefficients of f . Hence writing f ( x ) = a + a x + · · · + a d x d , (8)and letting N denote the number of solutions to the congruence y ≡ f ( x ) (mod q ) , (9)with 1 x H, y H, (10)it is sufficient to show that N H /d ( d +1)+ o (1) q /d ( d +1) + H /d + o (1) . We define the lattice L = { ( z, w , . . . , w d ) ∈ Z d +1 : z + a w + · · · + a d w d ≡ q ) } , (11)the convex body D = { ( h , . . . , h d ) ∈ R d +1 : | h | dH and | h i | dH i for 1 i d } , (12)and let λ , . . . , λ d +1 denote the successive minima of L with respect to D . We distinguish two cases depending on whether λ d +1 < λ d +1 < . (13)Let s = d ( d + 1)2 , and suppose ( x , y ) , . . . , ( x s , y s ) are 2 s solutions to (9) satisfying (10).We have z ≡ f ( x ) + · · · + f ( x s ) − f ( x s +1 ) − · · · − f ( x s ) (mod q ) , (14)for some | z | sH. We define the set X by X = { x H : there exists 1 y H such that y ≡ f ( x ) (mod q ) } , so that |X | = N. (15)Let J ( w , . . . , w d ) denote the number of solutions to the system ofequations x j + · · · + x js − x jd +1 − · · · − x j s ≡ w j (mod q ) , j d, (16) B. KERR AND A. MOHAMMADI with variables satisfying x , . . . , x s ∈ X , and when w = · · · = w d = 0 , we write J (0 , . . . ,
0) = J. The equation (14) implies that N s X | z | sH X | w i | sH i z ≡ a w + ··· + a d w d (mod q ) J ( w , . . . , w d ) , and since J ( w , . . . , w d )= Z (0 , d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈X e πi ( α x + ··· + α d x d ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d e − πi ( α w + ··· + α d w d ) dα . . . dα d Z (0 , d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈X e πi ( α x + ··· + α d x d ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d dα . . . dα d = J, the above gives N s X | z | dH X | w i | dH i z ≡ a w + ··· + a d w d (mod q ) J |L ∩ D | J, (17)where L and D are given by (11) and (12). We recall that J denotesthe number of solutions to the system x j + · · · + x js = x jd +1 + · · · + x j s , j d, with x , . . . , x s ∈ X . By Lemma 9 we have J ≪ |X | s H o (1) = H o (1) N s , which by (17) implies that N s ≪ |L ∩ D | H o (1) . (18)Combining the assumption (13) with Lemma 5 and Lemma 6 gives |L ∩ D | ≪ λ . . . λ d +1 ≪ Vol( D )Vol( R d +1 / L ) . OINTS ON POLYNOMIAL CURVES IN SMALL BOXES 9
Since Vol( D ) ≪ H ( d + d +2) / and Vol( R d +1 / L ) = q, we see that |L ∩ D | ≪ H ( d + d +2) / q , and hence by (18) N ≪ H o (1) H ( d + d +2) / q ! /s = H /d ( d +1)+ o (1) q /d ( d +1) . (19)Consider next when λ d +1 > . (20)The dual lattice L ∗ and the dual body D ∗ are given by L ∗ = 1 q { ( z, w , . . . , w d ) ∈ Z : ∃ n ∈ Z such that n ≡ z (mod q ) , a i n ≡ w i (mod q ) } , and D ∗ = { ( h , h , . . . , h d ) ∈ R : sH | h | + d X i =1 sH i | h i | } . If λ ∗ denotes the first successive minimum of L ∗ with respect to D ∗ then by (20) and Lemma 7 there exists some constant c depending onlyon d such that λ ∗ c. This implies that L ∗ ∩ cD ∗ = ∅ , and hence there exist z, w , . . . , w d , n ∈ Z such that n ≡ z (mod q ) , a i n ≡ w i (mod q ) , (21)and | z | ≪ qH , | w i | ≪ qH i . (22)We note that since ( a d , q ) = 1 we have w d = 0. Supposing ( x, y ) is asolution to (9) and recalling (8), we have ny ≡ a n + a nx + · · · + a d nx d (mod q ) , so that writing w for the least residue of a n (mod q ), an applicationof (21) gives w + w x + · · · + w d x d − zy = tq, for some t ∈ Z . By (22), the number of possible values for t is O (1)and for each such value of t we apply Theorem 1. This gives N H /d + o (1) , (23)and the result follows by combining (19) and (23).5. Proof of Theorem 4
Applying shifts as in the proof of Theorem 3, it suffices to estimatethe number of solutions to a congruence of the form(24) y − c y ≡ a x + a x + a x + a (mod q ) , x, y H, where ( a , q ) = 1. Let N denote the number of solutions to the con-gruence (24) and let X denote the set of x for which ( x, y ) satisfies (24)for some 1 y H , so that(25) N |X | . For j = 1 , , I j = [ − H j , H j ] , j = 1 , , , and consider the set S ⊆ I × I × I , of all triples(26) x = ( x + . . . + x , x + . . . + x , x + . . . + x ) , such that x i ∈ X . For each x we let I ( x ) count the number of solutionsto the equation (26) with variables x , . . . , x ∈ X . Thus X x ∈S I ( x ) = |X | , and X x ∈S I ( x ) , is bounded by the number of solutions to the system of equations(27) x j + . . . + x j ≡ x j + . . . + x j , j = 1 , , , with variables x , . . . , x ∈ X . By Lemma 9 X x ∈S I ( x ) H o (1) |X | , OINTS ON POLYNOMIAL CURVES IN SMALL BOXES 11 and by the Cauchy-Schwarz inequality, we have X x ∈S I ( x ) |S| X x ∈S I ( x ) ! / , which implies that |S| > |X | H o (1) . Hence, there exist at least |X | H o (1) triples( x , x , x ) ∈ I × I × I , such that a x + a x + a x ≡ y − c y (mod q ) , for some y ∈ I and y ∈ I . In particular, we have that the congruence a x + a x + a x + c y + y ≡ q ) , ( x , x , x , y , y ) ∈ I × I × I × I × I , has a set of solutions S with(28) |S| > |X | H o (1) . We define the lattice L = { ( x , x , x , y , y ) ∈ Z : a x + a x + a x + c y + y ≡ q ) } , and the body D = { ( x , x , x , y , y ) ∈ R : | x | , | y | H, | x | , | y | H , | x | H } . It follows from (28) that |X | ≪ | Γ ∩ D | H o (1) . (29)Let λ i denote the i -th successive minimum of L with respect to D . Weconsider two cases depending on whether λ λ
1. By Lemma 5 and Lemma 6 | Γ ∩ D | ≪ H q , and hence by (29) |X | H / o (1) q / , (30)and the result follows from (25). Suppose next that λ > . (31) The dual lattice L ∗ and dual body D ∗ are given by L ∗ = 1 q { ( w , w , w , z , z ) ∈ Z : ∃ n ∈ Z such that w i ≡ a i n (mod q ) , i = 1 , , ,z ≡ c n (mod q ) , and z ≡ n (mod q ) } . and D ∗ = { ( w , w , w , z , z ) : X i =1 H i | w i | + 6 H | z | + 6 H | z | } . If λ ∗ denotes the first successive minimum of L ∗ with respect to D ∗ ,by Lemma 7 we have λ λ ∗ ≪ , and hence by (31) λ ∗ ≪ . This implies that there exists some integer n satisfying n ≪ qH , (32)and a i n ≡ w i (mod q ) , i = 1 , , c n ≡ z (mod q ) , for some w , w , w and z satisfying w i ≪ qH i , i = 1 , , , z ≪ qH . (33)For any solution ( x, y ) to (24) we have ny − z y = w x + w x + w x + qt, for some t ∈ Z . By (32) and (33) there are O (1) possible values of t and for each such value of t we apply Theorem 1. This implies thatthe number of solutions to the congruence (24) is bounded by N H / o (1) , and the result follows combining the above with (25) and (30). References [1] W. Banaszczyk,
Inequalities for convex bodies and polar reciprocal lattices in R n , Discrete Comput. Geom., (1995), 217–231.[2] E. Bombieri and J. Pila, The number of integral points on arcs and ovals , DukeMath. J., (1989), 337–357.[3] J. Bourgain and M.-C. Chang, On a multilinear character sum of Burgess , C.R. Acad. Sci. Paris, Ser. I, (2010), 115–120.
OINTS ON POLYNOMIAL CURVES IN SMALL BOXES 13 [4] J. Bourgain, C. Demeter and L. Guth,
Proof of the main conjecture in Vino-gradov’s mean value theorem for degrees higher than three , Ann. Math., (2016), 633–682.[5] J. Bourgain and M. Z. Garaev,
Sumsets of reciprocals in prime fields andmultilinear Kloosterman sums , Izv. Ross. Akad. Nauk Ser. Mat., (2014),19–72 (in Russian); translation in Izv. Math, (2014), 656–707.[6] J. Bourgain and M. Z. Garaev, Kloosterman sums in residue rings , Acta Arith., (2014), 43–64.[7] J. Bourgain, M. Z. Garaev, S. V. Konyagin and I. E. Shparlinski,
On congru-ences with products of variables from short intervals and applications , Proc.Steklov Math. Inst., (2013), 61–90.[8] T. H. Chan and I. E. Shparlinski,
On the concentration of points on modularhyperbolas and exponential curves , Acta Arith., (2010), 59–66.[9] M.-C. Chang,
Sparsity of the intersection of polynomial images of an interval ,Acta Arith., (2014), 243–249.[10] M.-C. Chang, J. Cilleruelo, M. Z. Garaev, J. Hernndez and I. E. Shparlin-ski and A. Zumalac´arregui,
Points on curves in small boxes and applications ,Michigan Mathematical Journal (2014), 503–534.[11] J. Cilleruelo and M. Z. Garaev, Concentration of points on two and threedimensional modular hyperbolas and applications , Geom. and Func. Anal., (2011), 892–904.[12] J. Cilleruelo and I. E. Shparlinski, Concentration of points on curves in finitefields , Monatsh. Math., (2013), 315–327.[13] J. Cilleruelo, M. Z. Garaev, A. Ostafe and I. E. Shparlinski,
On the concentra-tion of points of polynomial maps and applications , Math. Zeit., (2012),825–837.[14] B. Kerr,
Solutions to polynomial congruences in well shaped sets , Bull. Aust.Math. Soc., (2013), 435–447.[15] S. V. Konyagin, Estimates of character sums in finite fields , MathematicalNotes, (2010), 503–515.[16] T. Tao and V. Vu, Additive Combinatorics, Cambridge
Stud. Adv. Math. 105,Cambridge Univ. Press, Cambridge, 2006.[17] T. D. Wooley,
Nested efficient congruencing and relatives of Vinogradov’s meanvalue theorem , arXiv:1708.01220[18] Z. Zheng,
The distribution of zeros of an irreducible curve over a finite field ,J. Number Theory, (1996), 106–118.[19] A. Zumalac´arregui, Concentration of points on modular quadratic forms , In-tern. J. Number Theory, (2011), 1835–1839. Department of Pure Mathematics, University of New South Wales,Sydney, NSW 2052, Australia
E-mail address : [email protected] School of Mathematics and Statistics, University of Sydney, NSW2006, Australia
E-mail address ::