Expansions of quadratic maps in prime fields

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1 Expansions of quadratic maps in prime fields Mei-Chu Chang Department of Mathematics University of California, Riverside Abstract Let f(x) = ax 2 +bx+c Z[x] be a quadratic polynomial with a 0 mod p. Take z F p and let O z = {f i (z)} i Z + be the orbit of z under f, where f i (z) = f(f i (z)) and f 0 (z) = z. For M < O z, We study the diameter of the partial orbit O M = {z, f(z), f 2 (z),..., f M (z)} and prove that there exists c > 0 such that diam O M min { Mp c, For a complete orbit C, we prove that where T is the period of the orbit. log p M 4 5 p 5, M 3 log log M }. diam C min{p 5c, e T/4 }, Introduction. This paper belongs to the general theme of dynamical systems over finite fields. Let p be a prime and F p the finite field of p elements, represented by 2000 Mathematics Subject Classification.Primary B50, 37A45, B75; Secondary T23, 37F0 G99. Key words. dynamical system, orbits, additive combinatorics, exponential sums. Research partially financed by the National Science Foundation.

2 the set {0,,..., p }. Let f F p [x] be a polynomial, which we view as a transformation of F p. Thus if z F p is some element, we consider its orbit z 0 = z, z n+ = f(z n ), n = 0,,..., (0.) which eventually becomes periodic. The period T z = T is the smallest integer satisfying {z n : n = 0,,..., T } = {z n : n N} (0.2) We are interested in the metrical properties of orbits and partial orbits. More precisely, for M < T z, we define diam O M = max 0 n<m z n z. (0.3) Following the papers [GS] and [CGOS], we study the expansion properties of f, in the sense of establishing lower bounds on diam O M. Obviously, if M T, then diam O M M. But, assuming that f is nonlinear and M = o(p), one reasonably expects that the diameter of the partial orbit is much larger. Results along these lines were obtained in [GS] under the additional assumption that M > p 2 +ϵ. In this situation, Weil s theorem on exponential sums permits proving equidistribution of the partial orbit. For M p /2, Weil s theorem becomes inapplicable and lower bounds on diam O M based on Vinogradov s theorem were established in [CGOS]. Our paper is a contribution of this line of research. We restrict ourselves to quadratic polynomials, though certainly the methods can be generalized. (See [CCGHSZ] for a genaralization of Proposition 2 and Theorem 2 to higher degree polynomials and rational maps.) Our first result is the following. Theorem. There is a constant c > 0 such that if f(x) = ax 2 +bx+c Z[x] with (a, p) =, then with above notation, for any z F p and M T z, { diam O M min Mp c, log p M 4 5 p 5, M 3 }. log log M (0.4) In view of Theorem 2, one could at least expect that diam O M min(p c, e cm ) as is the case when M = T z. h g, if there exist constants C, M such that h(x) Cg(x) for all x > M. 2

3 In the proof, we distinguish the cases diam O M > p c 0 and diam O M p c 0, where c 0 > 0 is a suitable constant. First, we exploit again exponential sum techniques (though, from the analytical side, our approach differs from [CGOS] and exploits a specific multilinear setup of the problem). More precisely, Proposition in states that (for M T large enough) diam O M log p min ( M 5 4, M 4 5 p 5 ). (0.5) (Note that (??) is a clear improvement over Theorem 8 from [CGOS] for the case d = 2.) When diam O M p c 0, a different approach becomes available as explained in Proposition 2. In this situation, we are able to replace the (mod p) iteration by a similar problem in the field C of complex numbers, for an appropriate quadratic polynomial F (z) Q[z]. Elementary arithmetic permits us to prove then that log diam O M is at least as large as log M log log M. 3 Interestingly, assuming C a complete periodic cycle and diam C < p c 0, the transfer argument from Proposition 2 enables us to invoke bounds on the number of rational pre-periodic points of a quadratic map, for instance the results from R. Benedetto [B]. The conclusion is the following. Theorem 2. There is a constant c 0 > 0 such that if f(x) = ax 2 +bx+c Z[x] with (a, p) = and C F p is a periodic cycle for f of length T, then diam C min{p c 0, e T/4 }. (0.6) Diameter of Partial Orbits. Let f(x) = ax 2 + bx + c Z[x], where a 0 (mod p). Fix x 0 F p and denote the orbit of x 0 by O x0 = {f j (x 0 )} j Z +, where f j (x 0 ) = f(f j (x 0 )) and f 0 (x 0 ) = x 0. The period of the orbit of x 0 under f is denoted T = T x0 = O x0. For A F p, we denote the diameter of A by diam A = max p x y x,y A p, 3

4 where a is the distance from a to the nearest integer. We are interested in the expansion of part of an orbit. Proposition. For M < T, consider a partial orbit Then O M = {x 0, f(x 0 ), f 2 (x 0 ),..., f M (x 0 )}. diam O M log p min(m 5/4, M 4/5 p /5 ). (.) Proof. Let M = diam O M. Take I F p with I = M and O M I, then f(i) I M. (.2) We will express (??) using exponential sums. Let 0 φ be a smooth function on F p such that φ = on I and supp φ Ĩ, where Ĩ is an interval with the same center and double the length of I. Equation (??) implies that φ(f(x)) M x I and expanding φ in Fourier gives φ(x) = φ(ξ)e p (xξ), with φ(ξ) = φ(x)e p ( xξ). p ξ F p x F p Combining these gives φ(ξ) e p (ξf(x)) M. (.3) ξ F p x I We will estimate x I e p(ξf(x)) using van der Corput-Weyl. Take M 0 = O(M), e.g. M 0 = M. Then 00 e p (ξf(x)) M 0 e p (ξ ( a(x + y) 2 + b(x + y) )) + O(M 0 ) x I [ M0 0 y<m 0 = M0 0 y<m 0 x I ( e p ξ(ax 2 + 2axy + bx) ) ] 2 /2 + O(M 0 ) x I ) e p (ξ(x /2 x 2 )(a(x + x 2 ) + 2ay + b) + O(M 0 ). 0 y<m 0 x,x 2 I (.4) 4

5 (The second inequality is by Cauchy-Schwarz.) Take φ sufficiently smooth as to ensure that Equations (??) and (??) imply φ(ξ) ξ F p 0 y<m 0 x,x 2 I ξ F p φ(ξ) = O(). (.5) e p (ξ(x x 2 )(a(x + x 2 ) + 2ay + b)) /2 M 3/2. Hence by Cauchy-Schwarz and (??), φ(ξ) e p (ξ(x x 2 )(a(x + x 2 ) + 2ay + b)) M 3. (.6) ξ F p 0 y<m 0 x,x 2 I Fix x + x 2 = s 2M ; then φ(ξ) e p (ξ(2x s)(as + 2ay + b)) M 3. (.7) 2M ξ F p Next, for z F p, denote 0 y<m 0 x I η(z) = {(x, y) I [, M 0 ] : (2x s)(2ay + b + as) z (mod p)}, (.8) and write the left hand side of (??) as φ(ξ) η(z) e p (ξz) ξ F p z F p ( ) /2 ( φ(ξ) 2 η(z)e p (ξz) 2 ) /2 ξ F p ξ F p z F p (.9) ( ) /2 = φ(x) 2 ( ) /2 p η(z) 2 p x F p z F p ( ) /2, < 2M /2 η(z) 2 z F p 5

6 by Cauchy-Schwarz and Parseval. Recall that (a, p) =. Let I = I s, 2 I = [, M 0 ] + b+as 2a η(z) 2 = E(I, I ), z F p F p so that the multiplicative energy of I and I. It is well-known that } E(I, I ) log p max { I I, I 2 I 2 p log p max {M M, M } 2 M 2. p (.0) Thus, by (??), (??) and (??), { M 3 M /2 (log p) /2 max M /2 M /2, M } M. (.) M p /2 Distinguish the cases M M p and M M > p, and (??) implies { ( ) /4M M min log p 5/4, ( log p ) } /5 M 4/5 p /5. (.2) 2 Partial orbits of small diameters. For M < p c 0, one obtains the following stronger result. (Notations are as in Proposition.) Proposition 2. There exists c 0 > 0 such that ( ) diam O M > min p c 0, M 3 log log M. (2.) Consequently, diam O M min { Mp c 0 5, log p M 4 5 p 5, M 3 log log M }. (2.2) 6

7 Proof. Let O M = {x 0, x,..., x M } with x j = f(x j ) as before, and let diam O M = M. Since x j x 0 M, we can write x j = x 0 + z j with z j [ M, M ]. Thus, a, b, c, x 0 satisfy the M equations a(x 0 + z j ) 2 + b(x 0 + z j ) + c x 0 + z j+ (mod p), j = 0,..., M 2, and the F p -variety V p = M 2 j=0 [ (r + zj ) 2 + v(r + z j ) + w = u(r + z j+ ) (mod p) in the variables (u, v, w, r) F 4 p is therefore nonempty. Note that the coefficients of the M defining polynomials in Z[u, v, w, r] are O(M 2 ). Assume M < p c 0 (2.3) with c 0 > 0 small enough. Elimination theory 2 implies that V p as a C-variety. Hence there are U, V, W, R C such that for all j (R + z j ) 2 + V (R + z j ) + W = U(R + z j+ ), j = 0,..., M 2. Obviously, U 0, since z,..., z M 2 are distinct. We therefore have a quadratic polynomial satisfying F (z) := U (R + z)2 + V U (R + z) + W U R = : Az2 + Bz + C, (2.4) F (z j ) = z j+ in C, for j = 0,..., M 2. (2.5) Since z 0 = 0, (??) and (??) imply C = z Z [ M, M ] and the equations z 2 A + z B = z 2 z z 2 2A + z 2 B = z 3 z imply A, B Q with A = a d, B = b d, and a, b, d Z being O(M 3 ). Equation (??) becomes z j+ = a d z2 j + b d z j + C. (2.6) 2 See [C] where a similar elimination procedure was used in a combinatorial problem. In particular, see [C], Lemma 2.4 and its proof. 7

8 Hence a d z j+ + b ( a 2d = d z j + b ) 2 + C a 2d d b2 4d + b 2 2d. Putting and gives y j = a d z j + b 2d Z, j = 0,..., M 2d r s = C a d b2 4d 2 + b 2d with s > 0, (r, s) =, r, s = O(M 6 ), y j+ = yj 2 + r, j = 0,..., M 2. (2.7) s Next, write y j = α j /β j, where β j 2d and (α j, β j ) = ; thus (??) gives Note also that α j+ β j+ = α2 j β 2 j Write the prime factorizations + r, j = 0,..., M 2. (2.8) s α j = O(M 4 ). (2.9) s = p p v(p) and β j = p p v j(p), j = 0,..., M. Claim. 2v j (p) v(p), for j < M O(log log M ). Proof. We may assume v j (p) > 0. Case. 2v j (p) > v j+ (p). Fact 2.(which will be stated at the end of this section) and (??) imply that v(p) = 2v j (p). Case 2. 2v j (p) v j+ (p). Again, we separate two cases. Case 2.. 2v j+ (p) > v j+2 (p). Reasoning as in Case, we have v(p) = 2v j+ (p) 2 2 v j (p) > 2v j (p). 8

9 Case v j+ (p) v j+2 (p). Therefore, v j+2 (p) 2 2 v j (p). We repeat the argument for Case 2 with j = j +. Continuing this process, after τ steps, we obtain either v(p) 2v j (p) or v j+τ (p) 2 τ v j (p), (2.0) when necessarily τ log v j+τ (p) log log β j+τ log log d log log M. Since j + τ M, the claim is proved. It follows from the claim that β 2 j s for j < M O(log log M ). Back to (??), if v(p) > 2v j (p) for some j < M O(log log M ), then v j+ (p) = v(p). This contradicts to that β 2 j+ s. So we conclude that Hence which implies β 2 j = s =: s 2 for j < M O(log log M ). (2.) α j+ = α2 j s + r s, (2.2) α 2 j + r 0 mod s. (2.3) Let s = p v i i. Then α j satisfies (??) if and only if α j satisfies αj 2 + r 0 mod p v i i for all i. Since r is a quadratic residue modulo p v if and only if it is a quadratic residue modulo p for odd prime p, we have {π s (α j )} j 2 2 ω(s ) < e log s log log s < e 4 log M log log M. (2.4) Here π s (α j ) is the projection of α j in Z s. To show M > M 3 log log M, we assume log M < log M log log M. (2.5) 3 Then (??) implies there exists ξ Z s such that { J = 0 j M 2 : π s (α j ) = ξ} > M /2. (2.6) Thus α j α j2 s Z, for j, j 2 J, 9

10 and α j α j2 s, for j j 2 J. In particular there exists j J such that α j M /2 8 s and αj r /2 > M /2 8 s. (2.7) Claim. Either α j > 0 r /2 or α j+ > 0 r /2. Proof. Assume α j, α j+ < 0 r /2. (2.8) Hence, r /2 M /2 s by (??). From (??), (??) and (??) 0 r /2 s > α j+ s = α 2 j + r ( α j + r /2 )( α j r /2 ) r /2 M /2 a contradiction, proving the claim. Thus, there exists j < M/2 such that either or 8 s α j > 0s and α j > 0 r /2 (2.9) α j > 0s and α j+ > 0 r 2. (2.20) Clearly, (??) implies (??). Indeed, by (??), Iteration shows that α j+ s α 2 j r 99 00s α 2 j > 2 α j. M αj+ M > 2 3 αj > 2 M 3 3 contradicting to (??). This proves (??). Combining Proposition and (??), we have (??). Fact 2.. Let a d, a 2 d 2, a 3 d 3 Q be rational numbers in lowest terms, and p v p(d i ) d i. If a d + a 2 d 2 + a 3 d 3 = 0 and v p (d ) v p (d 2 ) v p (d 3 ), then v p (d ) = v p (d 2 ). 0

11 3 Full cycles In this section, we will prove Theorem 2. Assume M = diam C < p c 0 with c 0 as in Proposition??. The proof of Proposition?? gives a quadratic polynomial (cf. (??)) F (z) = z 2 + r s with r, s Z, s = O(M 6 ) (3.) and a rational F -cycle {y j } 0 j<t, i.e. and y j+ = F (y j ) for 0 j T 2 F (y T ) = y 0. We now invoke a result of R. Benedetto [B], which gives quantitative bounds on the number of preperiodic points of a polynomial f in a number field. ( z is preperiodic, if the set { z, f(z), f ( f(z) ),... } is finite. ) According to Theorem 7. in [B], the number of preperiodic points of F in Q is bounded by (2σ + ) [log 2 (2σ + ) + log 2 (log 2 (2σ + ) ) + 2] (3.2) with σ the number of primes where F has bad reduction. Hence σ ω(s) and (??) implies log M log log M T < 4 log M = 4 log diam C. (3.3) Acknowledgement. The author would like to thank the referee for many helpful comments and the mathematics department of University of California at Berkeley for hospitality. References [B] R. Benedetto, Preperiodic points of polynomials over global fields, J. Reine Angew. Math. 608 (2007), [CCGHSZ] M.-C. CHANG, J. CILLERUELO, M. GARAEV, J. HERNAN- DEZ, I. SHPARLINSKI, A. ZUMALACARREGUI POINTS ON CURVES IN SMALL BOXES AND APPLICATIONS (preprint).

12 [C] M.-C. Chang, Factorization in Generalized Arithmetic Progressions and Application to the Erdos-Szemeredi Sum-Product Problems, Geom. Funct. Anal. Vol. 3, (2003), [CGOS] J. Cilleruelo, M. Garaev, A. Ostafe, I. Shparlinski, On the concentration of points of polynomial maps and applications, Math. Zeit., (to appear). [GS] J. Gutierrez, I. Shparlinski, Expansion of orbits of some dynamical systems over finite fields Bull. Aust. Math. Soc. 82 (200),