DECIMATIONS OF L-SEQUENCES AND PERMUTATIONS OF EVEN RESIDUES MOD P

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1 DECIMATIONS OF L-SEQUENCES AND PERMUTATIONS OF EVEN RESIDUES MOD P JEAN BOURGAIN, TODD COCHRANE, JENNIFER PAULHUS, AND CHRISTOPHER PINNER Abstract. Goresky and Klaer conjectured that for any rime > 13 and any l-sequence a based on, every air of allowable decimations of a is cyclically distinct. The conjecture is essentially equivalent to the statement that the maing A d, with (d, 1) = 1, A, is a ermutation of the even residues (mod ) if and only if d = 1 and A 1 (mod ), for > 13. We rove the conjecture for > , and establish it in a number of other secial cases such as when.0007 < d < Introduction Let be an odd rime, Z = Z/(), A, d integers with (d, 1) = 1, A and let E, O be the set of even and odd residues (mod ), E = {,, 6, 8,..., 1} Z, O = {1, 3, 5, 7,..., } Z. Let AE d = {A d : E} Z. Since (d, 1) = 1 the maing A d ermutes the elements of Z. Our interest is in determining when this maing is a ermutation of the elements of E, that is, AE d O is emty. It is trivially a ermutation when A = 1 and d = 1. It is also known to be a ermutation in the following cases (, A, d) = (5, 3, 3), (7, 1, 5), (11, 9, 3), (11, 3, 7), (11, 5, 9) and (13, 1, 5). Clearly, we may assume A < / and d < /. GK-Conjecture (Generalized Goresky-Klaer conjecture [9]) With the ecetion of the si cases listed above, if (d, 1) = 1, 0 < A < /, d < / and (A, d) (1, 1) then AE d O is nonemty. This conjecture is motivated by an (essentially) equivalent conjecture concerning binary l-sequences based on, sequences a = {a i } i of zeros and ones with a i ( i mod ) (mod ), (the arity of the least ositive residue of i (mod )), or some shift a t = {a i+t } i of a. These sequences are strictly eriodic with eriod 1 when is a rimitive root. If a is an l-sequence based on then an allowable decimation of a is a sequence of the tye = a d where i = a d i, and (d, 1) = 1. Two eriodic binary sequences a and b with the same eriod T are cyclically distinct if a t b for all shifts a t, 0 < t < T. The following conjecture imlies that l-sequences roduce large families of cyclically distinct sequences with ideal arithmetic cross-correlation. Date: February 16,

2 JEAN BOURGAIN, TODD COCHRANE, JENNIFER PAULHUS, AND CHRISTOPHER PINNER Original GK-Conjecture. (Goresky and Klaer [9]) If > 13 is a rime, a rimitive root modulo, and a an l-sequence based on, then every air of allowable decimations of a is cyclically distinct. To see how this conjecture is related to the first one, notice that the sequence a is a cyclic ermutation of a d if and only if there is some A Z such that (A id mod ) ( i mod ) (mod ) for all i. If is a rimitive root then i runs through all nonzero residues (mod ) and so the revious congruence is true if and only if (A d mod ) ( mod ) (mod ) for every, that is, AE d = E. The assumtion that is a rimitive root modulo is essential for the connection with l-sequences but we believe this assumtion to be unnecessary for the validity of the first conjecture. The conjecture is elementary when d = 1; see the remark at the end of Section 6. Klaer, using a comuter, has verified the generalized conjecture for all rimes less than two million. Goresky, Klaer and Murty [10] roved the conjecture for d = 1 and for the case where 1 (mod ) and d = (+1)/. Goresky, Klaer, Murty and Sharlinski [11, Theorem.] sharening the work of [10], roved it for all values of d with (1) 0 < d ( 1) , or 0 > d ( 1) They also gave an uer bound on the number of ossible countereamles to the conjecture for a given. The main result of this aer is to establish that the conjecture is valid for all sufficiently large. To state our first theorem let () M = #{( 1,, 3, ) (Z ) : 1 + = 3 +, d 1 + d = d 3 + d }. Using the method of finite Fourier series and eonential sums we rove Theorem 1. If M < , then the GK-conjecture holds true. It is elementary (see [8, Lemma 3.]) that M < d( 1) for d > 0 and that M < 3 d ( 1) for d < 0 and thus we have the following imrovement of (1). Corollary 1. If.0007 < d < then the GK-conjecture holds true. A result analogous to Theorem 1 can be stated with M relaced by a binomial eonential sum bound. Let e ( ) denote the additive character on Z, e () = e πi/, and set 1 (3) Φ d = ma e (u,v) (0,0) (u + v d ), where u, v run through Z. =1 Theorem. If Φ d 7 9 then the GK-conjecture holds true. Unfortunately, the uer bound on M in Theorem 1 and the uer bound on Φ d in Theorem both fail if the quantity d 1 := (d 1, 1) is large, as shown in [7]. For small d 1 we are able to establish the desired uer bound on M for sufficiently large.

3 DECIMATION 3 Theorem 3. For any integer d with (d, 1) = 1, d 1 <.18( 1) 16/3, we have M /3. In Section 6, we use a different method involving multilicative characters to handle the case of large d 1. As it turns out, we are able to rove the Goresky- Klaer conjecture for d 1 sufficiently large. Theorem. a) If d 1 > 8( π log + 1), then the GK-conjecture holds true. b) If > and d 1 > 10 then the GK-conjecture holds true. It is a simle matter to deduce from Theorems 1, 3 and that the Goresky- Klaer conjecture is true for sufficiently large. Theorem 5. For any rime > the GK-conjecture holds true. Proof. If d 1.18( 1) 16/3 then by Theorem 3, M /3 < for The result then follows from Theorem 1. Otherwise d 1 >.18( 1) 16/3 > 10 for > , and so Theorem (b) yields the result. Remarks. 1. There are several available estimates for Φ d, such as the Weil bound (Φ d (d 1) ) or the Mordell bound (Φ d 1/ M 1/ ) but they (together with Theorem ) lead to weaker results than Corollary 1 and Theorem 5. The first author recently established a new tye of bound for a general eonential sum [, Theorem 1]. For the binomial of interest here (where (d, 1) = 1) it states that given ɛ > 0 there is a δ > 0 such that if d 1 < 1 ɛ then () Φ d < 1 δ. The roof of () uses additive combinatorics and harmonic analysis, and aeals to the Balog-Szemeredi-Gowers theorem; see [1]. It may not be easy to make the result numeric.. Wen-Feng Qi and Hong Xu [18] have roven the Goresky-Klaer conjecture for the case of odd rime owers e with e, e The methods develoed in this aer can be alied in the same manner to q-ary l-sequences, a i (q i (mod )) (mod q), where q is a rimitive root (mod ).. The methods can also be alied to the following generalization of a roblem of D.H. Lehmer. Given d relatively rime to 1 obtain an asymtotic formula for the number N d of even residues (mod ) such that d (mod ) is an odd residue. Our interest in the current aer is just establishing that N d is nonzero. In the classical Lehmer roblem d = 1 and it is well known (by the Kloosterman sum estimate) that N 1 /; see Zhang [], [3]. Many generalizations have aeared in recent years, Zhang [], [5], Alkan, Stan and Zaharescu [1], Cobeli and Zaharescu [], Sharlinski [19], Liu and Zhang [1], [15], [16], Louboutin, Rivat and Sarkozy [17], Yuan and Zhang [1], Xu and Zhang [0], among others. For general d we see that N d / for small d 1, but for d 1 large there can be bias. For eamle when d = + 3, N d /6. We will reort on this roblem and the roblem of q-ary l-sequences in a sequel. 5. Canetti, Friedlander & Sharlinski [3] have reviously shown a bound slightly stronger than Therem 3 for the average value of M (averaged over all values of d): 1 1 M 7( 1) 11/ τ( 1). 1 d=1

4 JEAN BOURGAIN, TODD COCHRANE, JENNIFER PAULHUS, AND CHRISTOPHER PINNER. Finite Fourier Series This section rovides a review of basic tools we shall need from the theory of finite Fourier series, and can be skied by the eerienced reader. Let be an odd rime, e ( ) = e πi / and = =1. Any comle valued function α defined on Z has a Fourier eansion α() = y a(y)e (y), where the coefficients a(y) are given by (5) a(y) = 1 α()e ( y). The convolution of two functions α and β on Z is defined by α β() = α(u)β(v) = α(u)β( u). u u v u+v= If α and β have Fourier eansions with coefficients a(y), b(y) resectively, then α β has coefficients a(y)b(y). Let I = {a + 1, a +,..., a + B} Z be an interval in Z with B, and χ I be the characteristic function of I with Fourier eansion χ I () = y a I(y)e (y). Then ( a I (0) = B/, a I (y) = 1 e ( a B 1 ) sin(πby/) )y sin(πy/), y 0, and (6) a I (y) = f(b, ) := 1 sin(πby/) sin(πy/), y where the summand is understood to be B when y = 0. In [5] the second author roved f(b, ) log + 1. π The main term in this uer bound cannot be imroved. Indeed, in [6, Equation 5] Cochrane and Peral showed f(b, ) = log + O(1). π Letting I = {1,,..., 1 } we see that χ E() = χ I ( 1 ) and so a E (y) = a I (y) and y a E(y) = y a I(y). Thus, (7) The same holds for y a O(y). Let y a E (y) log + 1. π I = {a 1 + 1, a 1 +,..., a 1 + B 1 }, J = {b 1 + 1,..., b 1 + B }, y

5 DECIMATION 5 be intervals of integers in Z with I = B 1, J = B and 1 B 1, B <, and let χ I, χ J have Fourier eansions χ I () = y a I (y)e (y), χ J () = y a J (y)e (y). The convolution χ I χ J, defined by χ I χ J () = u χ I(u)χ J ( u), has Fourier coefficients a I (y)a J (y). We make frequent aeal to Parseval s identity which states that if α is any comle valued function on Z with eansion α() = y a(y)e (y) then y a(y) = α(). The theory etends naturally to functions of two (or more) variables. If α(, y) is a comle valued function defined on Z then it has a finite Fourier eansion α(, y) = u a(u, v)e (u + vy). v In articular, if α(, y) = f()g(y) then the Fourier coefficients of α(, y) are just the roducts of the coefficients of f and g. 3. Proof of Theorem 1 To show there eists an E such that A d O, we must show there eists a solution (, y) to the equation A() d = y 1, over Z, with (, y) I 1 I where { I 1 = 0, 1,,..., 1 } Z, I = I 1 {0} Z. Put I = {0, 1,, 3,..., [( 1)/]} Z, J = {1,, 3,..., [( + 1)/]} Z, and let χ I, χ J be the characteristic functions of I, J with Fourier eansions χ I () = u a I (u)e (u), χ J () = v a J (v)e (v). Let α be the convolution α(, y) = χ I χ I () χ I χ J (y), with Fourier eansion α(, y) = u,v a(u, v)e (u + vy), where (8) a(u, v) = a I (u) a I (v)a J (v). In articular, (9) a(0, 0) = I 3 J.

6 6JEAN BOURGAIN, TODD COCHRANE, JENNIFER PAULHUS, AND CHRISTOPHER PINNER Since I + I I 1 and I + J I, α is suorted on I 1 I and so it suffices to show that A() d =y 1 α(, y) > 0. We have α(, y) = a(u, v)e (u + vy) A() d =y 1 0 say. Now, by (9), A() d =y 1 0 u,v = a(0, 0)( 1) + = Main + Error, (u,v) (0,0) 1 a(u, v)e ( 1 ( v) e u + v(a d 1 d ) ) =1 (10) Main = a(0, 0)( 1) = 1 I 3 J. To estimate the error term we break it u as Error = E 1 + E + E 3, where E 1 is the sum over u = 0, v 0, E the sum over u 0, v = 0 and E 3 the sum over u 0, v 0. For E 1 and E the sum over is -1 since (d, 1) = 1. Thus (11) E 1 a(0, v) = a I (0) a I (v)a J (v) I I 1/ J 1/ v v by the Cauchy-Schwarz inequality and Parseval s identity, and (1) E u a(u, 0) = u a I (u) a I (0)a J (0) = I J. = I 5/ J 1/, For E 3 we use a variant from the argument of Konyagin and Sharlinski [13, Section 7]. By invariance under the grou action we have (13) E 3 a(u, v) e (u + va d 1 d ) u 0 v 0 0 = β(u, v ) e (u + v d ) u 0 v 0 0, where (1) β(u, v ) = 1 1 v a(u, A 1 d v ), 0 and A 1 A d 1 1 (mod ). Net, from Hölder s inequality E e (u + v d ) β(u, v ) β(u, v ) u (15) = E 1/ E 1/ 5 E 1/ 6, say. Clearly, u 0 v 0 (16) E = M, u 0 v 0 1

7 DECIMATION 7 with M as in (). Net, using (8) E 5 = β(u, v ) = 1 a(u, A 1 d v ) = a(u, v) 1 u 0 v 0 0 u 0 v 0 u 0 v 0 = a I (u) a I (v) a J (v) u 0 v 0 1 a I (u) a I (v) a J (v) u 0 and so by Parseval s identity v 0 (17) E 5 ( I ) 3/ I 3/ ( J ) 1/ J 1/ ( I ) I. Finally, for E 6 we have E 6 = β(u, v ) = u 0 v 0 = 1 1 = 1 1 = 1 1 ( 1) 1 u 1,u,v 1,v < (v 1/v ) (u 1/u ) d (mod ) 1 u 1,u,j< 1 u 1,u < v 0 0 y 0 u 0 v 0 a(u 1, ju d 1) a(u, ju d ) a I (u 1 ) a I (u ) 1 a(u, A 1 d v ) a(yu, A 1 y d v ) a(u 1, v 1 ) a(u, v ) 1 j< a I (u) a I (j)a J (j) 1 u 0 j 0 = 1 1 I ( I ) a I (j)a J (j). j 0, ai (ju d 1)a J (ju d 1) ai (ju d )a J (ju d ) To evaluate the latter sum we aly Parseval s identity to α() = χ I χ J to obtain, a I (j)a J (j) = a I (j)a J (j) a I (0)a J (0) j 0 j = 1 3 α () 1 I J. If 1 (mod ) then I = +3 1, J = and the function α is given by, for 1 1, 1 α() = + 1, for 1 < 1, 0, otherwise, since the solutions to u + v = with (u, v) I J are (0, ), (1, 1), (, ),...,( 1, 1) in the first case and (u, v) = ( 1, 1 1 ), ( + 1, 1

8 8JEAN BOURGAIN, TODD COCHRANE, JENNIFER PAULHUS, AND CHRISTOPHER PINNER 1),..., ( 1, 1 ), in the second case. Thus α () = [ (( 1)/) ] = 1 96 ( 1)( + 3). If 3 (mod ) then I = J = +1 and, for 1 +1, 1 α() = + 1, for +1 < 1, 0, otherwise, and so α () = [ (( 3)/) ] +((+1)/) = 1 ( + 96 ( 3)( 1) 1)+. 16 Hence (18) a I (j)a J (j) = j 0 ( ( ) ) , if 1 (mod ),, if 3 (mod ). Let γ denote the value on the right-hand side of (18) and note that for > 10 6, γ Then (19) E 6 γ 1 I ( I ). By (15), (16), (17) and (19) we have (0) E 3 γ 1/ and then by (11) and (1), (1) Error I 5/ J 1/ M 1/ 1/ ( 1) 1/ I 3/ ( I ) 3/, + I J + γ 1/ If 3 (mod ), so that I = J = +1, then Error 1 ( + 1) γ1/ 6 M 1/ 1/ ( 1) 1/ I 3/ ( I ) 3/. M 1/ 1/ ( 1) 1/ ( + 1)3/ (3 1) 3/ while Main = 1 ( + 1) ( 1) 56. If > 10 6 and M < one can check with a calculator that Error < Main. A similar calculation can be made for the case 1 (mod ).

9 DECIMATION 9. Proof of Theorem The roof is almost identical to the roof of Theorem 1, the only change being in the estimate of E 3. We have by (13) E 3 a(u, v) e (u + va d 1 d ) u 0 v 0 0 Φ d a(u, v) u 0 v 0 ) and so () ( ) ( Φ d a I (u) a I (0) a I (v)a J (v) a I (0)a J (0) u v ( ) ( I Φ d I I 1/ J 1/ I J ), Error E 1 + E + E 3 I 5 J 1 + I J + Φ d I J ( I )( I J ). The main term is again Main = 1 I 3 J. With a calculator one can check that Error < Main rovided that Φ d 7 9 and > Proof of Theorem 3 For any integers k, l let M(k, l) denote the number of solutions in (Z ) of the system k 1 + k = k 3 + k l 1 + l = l 3 + l. We have the elementary bounds ([8, Lemma 3.]) { kl( 1), for 1 l < k < 1, (3) M(k, l) 3k l ( 1), for l < 0, l k, k + l < 1. Also, since 1 1 (mod ) for Z, we have M(k, l) = M(k, l ) for k k and l l (mod 1). Lemma 1. For any integers k, l, m we have M(k, l) M(mk, ml). Proof. For any nonzero A 1, A, A 3, A, B 1, B, B 3, B Z let M(k, l, A 1, A, A 3, A, B 1, B, B 3, B ) be the number of solutions in (Z ) of the system A 1 k 1 + A k = A 3 k 3 + A k B 1 l 1 + B l = B 3 l 3 + B l. We first note that for any choice of A i, B j, M(k, l, A 1, A, A 3, A, B 1, B, B 3, B ) M(k, l).

10 10JEAN BOURGAIN, TODD COCHRANE, JENNIFER PAULHUS, AND CHRISTOPHER PINNER Indeed, M(k, l, A 1, A, A 3, A, B 1, B, B 3, B ) is just e (α(a 1 k 1 + A k A 3 k 3 A k ) + β(b 1 l 1 + B l B 3 l 3 B l )) α,β e (αa i k i + βb i l i) e ( αa i k i βb i l i) α,β i=1 i 0 i=3 i 0 1/ e (αa i k i + βb i l i) 1/ e ( αa i k i βb i l i) i=1 α,β = M(k, l). i 0 Net, set m 1 = (m, 1) and let {w 1,..., w m1 } be a set of reresentatives for Z /(Z ) m. Then decomosing Z as a union over the different cosets of Z m, we see that M(k, l) = 1 m 1 1 m 1 m 1 m 1 m 1 m 1 i 1=1 i =1 i 3=1 i =1 m 1 m 1 m 1 m 1 i=3 α,β i 0 M(mk, ml, w k i 1, w k i, w k i 3, w k i, w l i 1, w l i, w l i 3, w l i ) M(mk, ml) = M(mk, ml). i 1=1 i =1 i 3=1 i =1 Lemma. If k l (mod 1) and either k or l is corime to 1 then M(k, l) 3. Proof. Suose without loss of generality that (l, 1) = 1. Let m satisfy ml 1 (mod 1) and ut d km (mod 1) with 1 < d <. Then M(k, l) = M(km, lm) = M(d, 1) d 3. Let () λ 1 = (l, k), λ = (l, k, 1), l + = l, l = l, (5) δ + = and (k l) (k + l), δ =, λ 1 λ 1 M + (k, l) = M(k, l) for 1 l < k < 1, M (k, l) = M(k, l) for 1 l < k, l + k < 1. The net lemma is essentially Corollary 3.1 of [7] with the imlied constants made elicit. Lemma 3. For 1 l k < 1 then for k < 1 3 ( 1) 3 λ l 1 6 ±, where M ± (k, l) λ ( 1) + k l ± ( 1) + ( 1) µ µ = ma{768 5 /3 kl ± δ 1 3 ± λ/λ 1, 557δ ± λ}.

11 DECIMATION 11 Proof. We follow the roof of Corollary 3.1 of [7]. For u = (u 1, u ) Z define C ± (u) = #{ Z From (.1) of [7] we have : k 1 = u 1 y k, ±l 1 = u y ±l for some y Z }. M ± (k, l) λ ( 1) + k l ± ( 1) + λ( 1) N C±(u i ), where u 1,..., u N reresent the N distinct non-emty sets of being counted as u varies, ordered so that C ± (u 1 ) C ± (u ) C ± (u N ) > 0. Thus, it suffices to show that λ N i=1 C ±(u i ) ( 1)µ. Let 9 ( 1) T = 7 ( 1) ( δ ± kl ±/λ 1 ) 3 i=1, if kl ± /λ 1 < 1 δ ±, ( δ ) ± kl ±/λ 1, if kl ± /λ 1 1 δ ±. When k < 1 3 ( 1) 3 λ l 1 6 ± it was shown in Lemma 3.1 of [7] that (6) C ± (u t ) 6 5 ( 1) 5 (kl± /λ 1 ) 3 5 t 5 δ 1 5 ±, 1 t T, in articular C ± (u T ) 37 5 δ ± when kl ± /λ 1 < 1 δ ±. If T = 0 then as shown at the end of the roof of [7, Lemma 3.1], we must have (kl ± /λ 1 ) 1 δ ± and thus from the definition of T, δ ± < 7 (kl ± /λ 1 ) 1 /( 1) 1. We can then use the trivial bounds C ± (u i ) min{ 1, kl ± /λ 1 } (kl ± /λ 1 ) 5 6 ( 1) 1 6, and N i=1 C ±(u i ) 1, to get N N λ C±(u i ) λ(kl ± /λ 1 ) ( 1) 6 C ± (u i ) i=1 λ(kl ± /λ 1 ) 5 7 µ 6 ( 1) 6 ( 1) Suose now that T > 0. Set L = δ 1 3 ± i=1 5 5/3 7 ( 1) (kl ± /λ 1 ) When L < T we have by (6) C±(u i ) 5/5 ( 1) /5 (kl ± /λ 1 ) 6/5 δ /5 ± and L<i N i L. i L 5/5 ( 1) /5 (kl ± /λ 1 ) 6/5 δ /5 ± 5L 1/5 9 5 /3 (kl ± /λ 1 )δ 1/3 ± ( 1), i /5 C ±(u i ) 6/5 ( 1) /5 (kl ± /λ 1 ) 3/5 δ 1/5 ± (L + 1) /5 L<i N 6/5 ( 1) /5 (kl ± /λ 1 ) 3/5 δ 1/5 ± (L + 1) /5 ( 1) 8 5 /3 (kl ± /λ 1 )δ 1/3 ± ( 1), C ± (u i )

12 1JEAN BOURGAIN, TODD COCHRANE, JENNIFER PAULHUS, AND CHRISTOPHER PINNER giving λ N i=1 C ±(u i ) /3 (λ/λ 1 )kl ± δ 1/3 ± ( 1). Plainly 5 5/3 7 ( 1 δ 3 ± δ ± 1) (kl is less than 1 ±/λ 1) 7 ( 1) (kl and less than 1 ±/λ 1) 9/ ( 1) (kl±/λ1)3/ when (kl ± /λ 1 ) 5 /3 3/5 δ /3 ±. Thus L < T unless (kl ± /λ 1 ) < 5 /3 3/5 δ /3 1 δ ± in which case C±(u i ) 5/5 ( 1) /5 (kl ± /λ 1 ) 6/5 δ /5 ± 5T 1/5 and i T T <i N 19/ 5( 1)(kl ± /λ 1 ) 3/ δ 1 ± 3/5 δ ± ( 1) C ±(u i ) 37/5 δ ± T <i N C ± (u i ) 37/5 δ ± ( 1), giving λ i N C ±(u i ) ( 3/5 + 37/5 )δ ± λ( 1) 557δ ± λ( 1). Theorem 3 is just a secial case of the following theorem with k = d, l = 1. δ 3 ± ± < Theorem 6. Let 1 l < k < 1 be ositive integers with (kl, 1) = 1, and for M (k, l), k + l < 1. Let d = (k l, 1), for M + (k, l), + for M (k, l). If d <.18( 1) 16/3 then M ± (k, l) /3. Proof. Let k, l be integers with l < k < 1 and (kl, 1) = 1. By Lemma the bound on M ± (k, l) is trivial if /3 and so we may assume that > The idea is to make a transformation of the tye m so that Lemma 3 can be effectively alied. Choose m so that (7) mk α mod ( 1), ±ml β mod ( 1), (lus sign for S + and minus for S ) with (8) 0 α ( 1) 3, β c( 1) 3, c = 60/3 5 /3 = , c (α, β) (0, 0). Such a air (α, β) eists since the set of all (α, β) satisfying (7) is a lattice of volume 1 (or one can aly Dirichlet s bo rincile.) Now, ( 1) m (since (α, β) (0, 0)) and so, since (lk, 1) = 1 we have α 0 and β 0. If α = β then 1 m(k l), 1 m and β ( 1)/d contradicting our assumtion d on the size of d. Thus α β. Set β = { β if β > 0, β if β < 0. Case i: Suose that α 100 β. Then by Lemma 1 and (3) we have, M ± (k, l) M(α, β) 3α β 300 β /3. Case ii: Suose that α > 100 β and α 5 ( 1) /3 λ 1/6 1 (β ) 1/6. Then (β ) 1/6 (3/c) /69, β (3/c) 6 /3. By Lemma 1 and (3) we get M ± (k, l) M(α, β) 3 αβ 3 c 16/3 ( 3 c ) 6 / /3.

13 DECIMATION 13 Case iii: Suose that α > 100 β and that α < 5 ( 1) /3 λ 1/6 1 (β ) 1/6, so that Lemma 3 alies. In articular, since δ ± = α β /λ 1 we have.99 α δ + α α, δ 1.01 α λ 1 λ 1 λ 1 λ 1 and β δ 1/3 ± β α 1/3 λ 1/3 1. The value µ in Lemma 3 is bounded by ma{768 5 /3 (λ/λ 1 )α β α 1/3 λ 1 1/3, 557(1.01α)} ma{ /3 α /3 β /3, 563α} ma{ /3 c /3 ( 1) 0/3, 563c 1 ( 1) 16/3 } ma{ ( 1) 0/3, 107( 1) 16/3 } = ( 1) 0/3. Thus we get M ± (k, l) M(α, β) (c 7/3 ) + (1/c) 39/ /3 9 60/ / / /3. 6. Proof of Theorem Let A, d be integers such that 0 < A < /, d < /, (A, d) (1, 1). Put d 1 = ( 1, d 1) and k = ( 1)/d 1. Let B be chosen so that B and AB d 1 1 (mod ) ; such a B eists since either d = 1, A 1, or d 1 and B d 1 takes on at least two distinct nonzero values (mod ). Put C AB d 1 (mod ) with / < C < /, C 0, 1. Suose that we can find an element of the form Bz k E such that BCz k O. Then A(Bz k ) d BCz k O, that is, AE d O is nonemty. Let Bz k (mod ), y BCz k (mod ). We count the number N of solutions of the congruence y C (mod ) such that E, B 1 is a k-th ower, and y O. Then letting ψ k =ψ 0 denote a sum over all multilicative characters ψ (mod ) satisfying ψ k = ψ 0, where ψ 0 is the rincial character, we have N = 1 ψ(b 1 ) χ E ()χ O (C) k ψ k =ψ 0 = 1 χ E ()χ O (C) + 1 ψ(b 1 )χ E ()χ O (C) k k (9) = Main + Error. ψ ψ 0 Main Term: Suose first that 1 < C < /. The main term is just the number of values of n {1,,..., 1 } such that (j 1) < nc < j for some j, that is (j 1) C < n < j C with 1 j [C/]. Thus, using [] [ y] [y], we have (30) Main = 1 k [C/] j=1 [ ] j C [ ] (j 1) 1 C k [C/] j=1 [ ] = 1 C k [ ] C [ ]. C We consider first a few small values of C. Let S denote the sum aearing in the main term, S = k(main). If C = then S = [/] [/] 1. If C = 3 then

14 1JEAN BOURGAIN, TODD COCHRANE, JENNIFER PAULHUS, AND CHRISTOPHER PINNER S = [/3] [/6] 1 6. For C = we have S = ([/] [/8]) + ([/] [3/8]) 3. For < C < we have For 5 C < / we have [C/] [/C] = [C/] C 1 [C/] [/C] C ( C C 1 C = + 3 ( + 1 C + C The quantity being subtracted takes on its maimum value when C = 1 and so we obtain [ ] C [ ] 1 C 8 1 > 3 8. Thus in all cases S ( 3)/8. Net assume that / < C 1. Then nc O if and only if nc is even and so we relace C with C and count the number of values n with j < nc < (j + 1) for some j with 0 j [(C 1)/]. Then, [(C 1)/] j=0 [ (j + 1) C ] [ ] j C [ C + 1 ) ). ] [ ], C and the lower bound follows as before. Thus we have uniformly, (31) Main 3 8k. Error Term: Let ψ be a nonrincial character (mod ). Then ψ(b 1 )χ E ()χ O (C) = ( a E (y)e (y))( a O (z)e (zc))ψ(b 1 ) y z = a E (y)a O (z)g(y + Cz, B 1 ), y z where G(y+Cz, B 1 ) is the Gauss sum G(y+Cz, B 1 ) = e ((y+cz))ψ(b 1 ), of modulus, unless y + Cz = 0 in which case it vanishes. Thus we obtain from (7) ψ(b 1 )χ E ()χ O (C) a E (y) ( ) a O (z) π log + 1, y z and ( ) Error (1 1/k) π log + 1. We conclude from (9) and (31) that N is ositive rovided that 3 8 (k 1)( π log + 1). If d 1 > 1 then k 1 and 3 k 1 1 k = d 1. Thus N is ositive rovided that d 1 > 8( π log + 1).

15 DECIMATION 15 To rove art (b) of the theorem suose that > and d 1 > 10. In [7, Proosition 1.1] we roved e (a d + b) d 1 + 3/, d 1 0 for any nonzero a, b. Thus Theorem can be alied if d 1 + 3/ d 1 either ( d 1 < 1 ) 7 ( 7) / or d 1 > 1 < 7 9. Otherwise, ( ) 7 ( 7) /. The first inequality fails for d 1 > 10 and > Thus the second inequality holds true. But for > , it imlies that d 1 > 8( π log() + 1). Thus art (a) of the theorem alies. Remark: When d = 1 there is no error term in the above calculation and we obtain that AE O > 3 8. Thus AE O is nonemty for any odd rime and A 1. Acknowledgement. The authors eress their thanks to the referee for his/her helful suggestions in imroving the organization of this aer and for ointing out valuable connections between this work and related roblems. References [1] E. Alkan, F. Stan and A. Zaharescu, Lehmer k-tules, (English summary) Proc. Amer. Math. Soc. 13 (006), no. 10, [] J. Bourgain, Mordell s eonential sum estimate revisited, J. Amer. Math. Soc. 18, (005), no., [3] R. Canetti, J. Friedlander and I. Sharlinski, On certain eonential sums and the distribution of Diffie-Helman triles, J. London Math. Soc. () 59 (1999), [] C. Cobeli and A. Zaharescu, Generalization of a roblem of Lehmer, Manuscrita Math. 10 (001), no. 3, [5] T. Cochrane, On a trigonometric inequality of Vinogradov, J. Number Theory 7, (1987), [6] T. Cochrane and J.C. Peral, An asymtotic formula for a trigonometric sum of Vinogradov, J. Number Theory 91 (001), [7] T. Cochrane and C. Pinner, Steanov s method alied to binomial eonential sums, Quart. J. Math. 5 (003), [8], An imroved Mordell tye bound for eonential sums, Proc. Amer. Math. Soc. 133 (005), no., [9] M. Goresky, A. Klaer, Arithmetic cross-correlations of FCSR sequences, IEEE Trans. Inform. Theory, 3 (1997), [10] M. Goresky, A. Klaer and R. Murty, On the distinctness of decimations of l-sequences, (Proceedings of SETA 01), T. Helleseth, ed., Discrete Mathematics and Comuter Science, Sringer Verlag, NY, 00. [11] M. Goresky, A. Klaer, R. Murty and I. Sharlinski, On decimations of l-sequences, SIAM J. Discrete Math. 18 (00), no. 1, (electronic). [1] T. Gowers, A new roof of Szemerédi s theorem for arithmetic rogressions of length, GAFA 8 (1998), no. 3, [13] S. Konyagin and I. Sharlinski, Character sums with eonential functions and their alications, Cambridge Univ. Press 136, United Kingdom, [1] H. Liu and W. Zhang, On a roblem of D. H. Lehmer, Acta Math. Sin. (Engl. Ser.) (006), no. 1,

16 16JEAN BOURGAIN, TODD COCHRANE, JENNIFER PAULHUS, AND CHRISTOPHER PINNER [15] Hybrid mean value results for a generalization on a roblem of D. H. Lehmer and hyer-kloosterman sums, Osaka J. Math. (007), no. 3, [16] Hybrid mean value for a generalization of a roblem of D. H. Lehmer, Acta Arith. 130 (007), no. 1, [17] S. Louboutin, J. Rivat and A. Sárközy, On a roblem of D. H. Lehmer, (English summary) Proc. Amer. Math. Soc. 135 (007), no., [18] W.F. Qi and H. Xu, Further results on the distinctness of decimations of l-sequences, IEEE Trans. on Info. Theory 5 (006), no. 8, [19] I. Sharlinski, On a Generalisation of a Lehmer Problem, Math. Zeit to aear (rerint (006), arxiv:math/06071v [math.nt]). [0] Z. Xu and W. Zhang, A roblem of D. H. Lehmer and its mean value, Math. Nachr. 81 (008), no., [1] Y. Yuan and W. Zhang, On the generalization of a roblem of D. H. Lehmer, Kyushu J. Math. 56 (00), no., [] W. Zhang, On a roblem of D. H. Lehmer and its generalization, Comositio Math. 86 (1993), no. 3, [3] A roblem of D. H. Lehmer and a generalization of it, (Chinese) J. Northwest Univ. 3 (1993), no., [] A roblem of D. H. Lehmer and its generalization. II, Comositio Math. 91 (199), no. 1, [5] On a roblem of D. H. Lehmer and Kloosterman sums, Monatsh. Math. 139 (003), no. 3, School of Mathematics, Institute of Advanced Study, Princeton, NJ 0850, USA address: Bourgain@math.ias.edu Deartment of Mathematics, Kansas State University, Manhattan, KS address: cochrane@math.ksu.edu Deartment of Mathematics, Kansas State University, Manhattan, KS address: aulhus@math.ksu.edu Deartment of Mathematics, Kansas State University, and Manhattan, KS address: inner@math.ksu.edu