aa r X i v : . [ m a t h . N T ] N ov A REMARK ON SOLUTIONS OF THE PELL EQUATION
J. BOURGAINA
BSTRACT . The main purpose of this Note is to provide a non-trivial bound oncertain Kloosterman sums as considered in the recent paper of E. Fouvry [F],leading to a small improvement of his result on the Pell equation.
0. INTRODUCTIONThis Note is motivated by the recent work of E. Fouvry [F] related to the size ofthe fundamental solution of the Pell equation and related conjectures due to Hooley[H]. Let us briefly recall the background. Let D be a non square positive integerand ε D the fundamental solution to the Pell equation t − Du = 1 . (0.1)Following [H] and [F], introduce for α > and x ≥ the set S f ( x, α ) = |{ ( ε D , D ); 2 ≤ D ≤ x, D non square and ε D ≤ D + α }| . (0.2)For < ε ≤ α ≤ , Hooley established the asymptotic formula S f ( x, α ) ∼ α π x (log x ) for x → ∞ (0.3)(see also Theorem A in [F]). For α > , he went on making several furtherconjectures on the size of this set, in particular the behavior. S f ( x, α ) ∼ B ( α ) x (log x ) (0.4)with B ( α ) = π (cid:16) α − (cid:17) for < α ≤ π (cid:16) α − (cid:17) + π ( α − for ≤ α ≤ π (cid:16) α − (cid:17) + π (cid:16) α − (cid:17) for α > . (0.5)Some of the heuristics in [H] was formalized in [F], where a lower bound for ≤ α ≤ S f ( x, α ) ≥ π (cid:0) α − − α ) − o (1) (cid:1) x (log x ) (0.6) is obtained. In the same paper, Fouvry also shows how to derive that S f ( x, α ) > (cid:0) B ( α ) − o (1) (cid:1) x (log x ) for ≤ α ≤ (0.7)assuming a rather modest (but still unproven) bound on certain Kloosterman sums.The approach in [F] leads indeed to exponential sums of the type X u ∼ U β u X u ∼ U e (cid:16) h ¯ u u (cid:17) (0.8)where ( u , u ) = 1 and ¯ u denotes the inverse of u mod u ; | β u | ≤ .Fouvry’s lower bound (0.6) relies essentially on various estimates on (0.8). Theinner sum is typically an incomplete Kloosterman sum for which presently non-trivial bounds (with power gain) are only available for U > U ε . Comparedwith (0.7), the weaker lower bound (0.6) is due to the exclusion of certain ranges of U , U for which (0.8) cannot be adequately estimated. In particular, when α > is near , further estimates on (0.8) for U < U < U ε become quite relevant.Following an approach initiated by Karacuba and developed further in [B-G1],[B-G2], it turns out that one can bound an incomplete Kloosterman sum X x
1. S
OME P RELIMINARY L EMMAS
Lemma 1.
Fix some r ∈ Z + and β > sufficiently small. For n ≤ N , let n = p p · · · , p ≥ p ≥ · · · be its prime factorization. Then there is a subset E ⊂ { , . . . , N } , | E | < β (cid:16) log 1 β (cid:17) r (1.1) such that for n < N, n E , n has at least r prime factors and p r > N β (1.2) Proof.
First, the set of integers n ≤ N with fewer than r prime factors is boundedby C r N (log log N ) r − log N . (1.3)We may also assume p ≥ N α , excluding a set of size ψ ( N, N α ) . α α N (1.4)where α > β is a parameter to be determined.Next, assume n of the form n = p · · · p r n ′ , r < r (1.5) p > N α , p , . . . , p r ≥ N β and n ′ with no factors larger than N β .Distinguishing the case n ′ < y and n ′ ≥ y ( y = N γ a parameter), the numberof those integers may be bounded by X n ′
Next statement is a variant of a Lemma due to Karacuba.
Lemma 2. (cid:12)(cid:12)(cid:12)n ( x , . . . , x ℓ ) ∈ P ℓ ; x i , x i + ℓ ∼ M i and x + · · · + 1 x ℓ = 1 x ℓ +1 + · · · + 1 x ℓ o(cid:12)(cid:12)(cid:12) < (2 ℓ ) ℓ ℓ Y i =1 M i log M i (1.8) Proof. If x + · · · + x ℓ = x ℓ +1 + · · · + x ℓ , then x i | Q ≤ j ≤ ℓj = i x j and hence x i ∈ { x j ; j = i } . (cid:3) The statement follows2. A B
OUND ON AN I NCOMPLETE K LOOSTERMAN S UM Our aim is to bound the incomplete Kloosterman sum X ≤ x ≤ N, ( a,q )=1 e q ( a ¯ x ) (2.1)where q ∈ Z + , ( a, q ) = 1 and ¯ x is the inverse of x ( mod q ) .The range N = q ρ for some fixed < ρ < .Let < β < be a parameter and apply Lemma 1 with r some fixed integer, tobe specified later. The integers x ∈ { , . . . , N }\ E admit a factorization x = p · · · p r x ′ with p ≥ · · · ≥ p r > p β , prime factors of x ′ are ≤ p r . (2.2)We require moreover that p i > (cid:16) N (cid:17) p i +1 for ≤ i ≤ r. (2.3)The size of the complementary set is indeed at most X pp ′
12) = X z ,...,z r µ ( z ) · · · µ r ( z r ) e q ( az . . . z r ) and µ i ( z ) = |{ ( x , . . . , x ℓ i ) ∈ ( I i ∩ P ) ℓ i ; ¯ x + · · · + ¯ x ℓ i − · · · − ¯ x ℓ i ≡ z ( mod q ) }| , From (2.11) and the preceding, Lemma 2 implies that k µ i k = X z µ i ( z ) < (4 ℓ ) ℓ ( ˜ M i ) ℓ i (2.13)while obviously k µ i k = ( ˜ M i ) ℓ i . (2.14)Note also that by (2.10), certainly ℓ i ≥ β i , hence ( ˜ M i ) ℓ i > q . (2.15)From (2.13), (2.14) max µ ( z ) = k µ k ∞ < (4 ℓ ) ℓ (cid:16) M i log M i (cid:17) ℓ < q − k µ k . The previous argument shows more generally that if q ≃ q ε and ≥ ε > β i (2.16)then max ξ ∈ Z /q Z X z ≡ ξ ( mod q ) µ ( z ) < q − k µ k . (2.17)In order to derive (2.17), take ≤ ℓ ≤ ℓ i such that β i (8 ℓ − < ε ≤ β i (8 ℓ + 6) and proceed as above with x ℓ +1 , . . . , x ℓ i , x ℓ i + ℓ +1 , · · · , x ℓ i frozen.Thus we define for z ∈ Z /q Z the density µ ′ ( z ) = |{ ( x , . . . , x ℓ ) ∈ ( I i ∩P ) ℓ ; ¯ x + · · · +¯ x ℓ − ¯ x ℓ +1 −· · ·− ¯ x ℓ ≡ z ( mod q ) }| J. BOURGAIN for which again max z ∈ Z /q Z µ ′ ( z ) < q − k µ ′ k holds. Since µ was disintegrated in such measures µ ′ , (2.17) follows.Our aim is to apply Theorem (**) from [B]. This result may be reformulated asfollows. Lemma 3.
Given γ > , there is ε = ε ( γ ) > , τ = τ ( γ ) > and k = k ( γ ) ∈ Z + such that the following holds.Let µ , . . . , µ k be probability densities on Z /q Z satisfying max ξ ∈ Z /q Z h X Z ≡ ξ ( mod q ) µ i ( z ) i < q − γ if q | q, q > q ε . (2.18) Then max a ∈ ( Z /q Z ) ∗ (cid:12)(cid:12)(cid:12) X e q ( ax . . . x k ) µ ( x ) · · · µ k ( x k ) (cid:12)(cid:12)(cid:12) < Cq − τ . (2.19)In view of (2.17) we may take γ = . Note that since P ri =1 β i ∼ ρ ≤ , wemay assume, up to reordering, that β , . . . , β [ r ] ≤ r < ε (cid:0) (cid:1) by taking r largeenough. Assume also r > k (cid:0) (cid:1) for Lemma 3 to apply.Exploiting ≤ i ≤ [ r ] , we can then satisfy (2.16) for ε ≥ ε ( ) . As a conse-quence of (2.19) one deduces that | (2 . | < Cq − τ ( ) k µ k . . . k µ r k . (2.20)Recalling (2.11) and (2.14), we proved that the sum S introduced in (2.8) satisfies | S | < ˜ M · · · ˜ M r q − τ ( )(2 r ℓ ...ℓ r ) − . (2.21)By (2.10), ℓ i ∼ β i < ρβ for i = 1 , . . . , r , where r is some constant. Therefore | S | < M · · · M r q − c ( ρβ ) C (2.22) | (2.6) | < N q − c ( ρβ ) C . (2.23)Finally, substitution in (2.7) gives (cid:12)(cid:12)(cid:12) X x ≤ N,x E, ( x,q )=1 e q ( a ¯ x ) (cid:12)(cid:12)(cid:12) . (log N ) C N − c ( ρβ ) C (2.24)where N = q ρ . In summary, we proved the following REMARK ON SOLUTIONS OF THE PELL EQUATION 9
Proposition 4.
Let N = q ρ . Given N < β < , there is a subset E ⊂{ , . . . , N } (independent of q ) satisfying | E | . β (cid:16) log 1 β (cid:17) C N (2.25) and such that for ( a, q ) = 1 (cid:12)(cid:12)(cid:12) X x ≤ N,x E, ( x,q )=1 e q ( a ¯ x ) (cid:12)(cid:12)(cid:12) . (log N ) C N − c ( ρβ ) C (2.26) with c, C > absolute constants.
3. F
OUVRY ’ S A PPROACH
Following [F], we introduce the sets (for α > S ( x, α ) = { ( η D , D ); 2 ≤ D ≤ x, D nonsquare, ε D ≤ η D ≤ D + α } (3.1)and S f ( x, α ) = { ( ε D , D ); 2 ≤ D ≤ x, D nonsquare, ε D ≤ D + α } (3.2)with ε D the fundamental solution of the Pell equation t − Du = 1 (3.3)and η D belonging to the set of solutions {± ε nD ; n ∈ Z } . (3.4)One has then (of [F], 17) S ( x, α ) = X ≤ u ≤ X α X Ω ∈R ( u ) X t ≡ Ω( mod u ) Y ( u,α ) ≤ t ≤ Y ( u ) (3.5)when R ( u ) = { Ω mod u ; Ω ≡ mod u ) } (3.6)and denoting X α = 12 ( x α − x − − α ) (3.7) Y ( u, α ) ∼ α u α , Y ( u ) = ux . (3.8)As noted in [F] S f ( x, α ) = S ( x, α ) for α ≤ (3.9) and S ( x, α ) = S f ( x, α ) + S (cid:16) x, α − (cid:17) for ≤ α ≤ . (3.10)Assume ≤ α ≤ . Then, following [F], §
4, we have S ( x, α ) ∼ L ( x, α ) + 1 π x (log x ) (3.11) L ( x, α ) = X X
2. For reasons of exposition, we first need to recall briefly some ofthe steps in Fouvry’s analysis, referring the reader to [F] for details.Assuming u , u ≥ coprime, set Φ( u , u ) = − ¯ u u + ¯ u u ( mod ˆ u ) (3.13)with u = u u , ¯ u ( resp. ¯ u ) the reciprocals ( mod u ) , ( resp. ( mod u ) .Arithmetical operations permit then to express L ( x, α ) as a sum ∞ X k =0 X ξ ∈R (2 k ) L ( x, α, ξ, k ) (3.14) L ( x, α, ξ, k ) = X u ,u X < k u u The corresponding main contribution is given by EM T ( x, U , U ) = X u ,u u ∼ U ,u ∼ U u ≡ ξ ,u = ξ ( u k )( u ,u , Y − Y u k u u . (3.18)A main portion of the analysis in [F] consists in bounding (3.16) in variousranges for U , U , retaining in (3.17) only those ranges for which conclusive esti-mates may be obtained. Summing the corresponding main terms (3.18) produces alower bound on L ( x, α ) leading to the minoration for ≤ α ≤ S ( x, α ) ≥ π (cid:16) (cid:16) α − (cid:17)(cid:16) − α (cid:17) − o (1) (cid:17) x (log x ) (3.19)as stated in Theorem 1 of [F].Using § U , U as well, hence narrowing further the excluded summation range. This leads to abetter minoration for α > close to .Let us be more precise. The conclusion from the analysis in § § ( u , u ) is the set A = { ( u , u ); u ≤ x , x u − ≤ u ≤ min( x α u − , x u ) } (3.20)leading to a contribution x X ( u ,u ) ∈A ( u ,u )=( u u , u u − o (cid:0) x (log x ) (cid:1) (3.21)to the main term.Let us consider the range u > x . Thus, recalling (3.17) x < u < x α u < u < min( x α u − , x u ) < x α − . (3.22)Returning to (3.16), we apply the Proposition from § q = u , N = U . Hence ρ ∼ . Let β > be a small parameter (to specify) and E ( U ) the exceptional set obtained. Thus from (2.20), (2.26) | E ( U ) | . β (cid:16) log 1 p (cid:17) C U (3.23) while for u ∼ U , h = 0 (cid:12)(cid:12)(cid:12) X u ∼ U u E e (cid:16) h ¯ u u (cid:17)(cid:12)(cid:12)(cid:12) < U (cid:0) ( h, u ) U − (cid:1) cβ C . (3.24)In the range (3.22), we only exclude the pairs ( u , u ) with u ∈ E ( U ) . Theircontribution in the main term is bounded by, cf. (3.21) O (cid:16) x X x
12 + ε X u ∼ U (cid:12)(cid:12)(cid:12) X u ∼ U u E ( U ) e (cid:16) h ¯ u u (cid:17)(cid:12)(cid:12)(cid:12) (3.26)with suitably restricted u variable in the inner sum. Hence (3.24) applies and weget (3.26) < x α − + ε U U ( x α − + ε U − ) cβ C < x α − (cid:0) x − α (cid:1) − cβ C (3.27)Hence we need to require α > close enough to and take β sufficiently large toensure that (3.27) < x . Thus β is at least a small power of α − .In conclusion, incorporating the above in Fouvry’s analysis leads to a smallimprovement in his Theorem 1 for α close to , to the extent of implying S f ( x, α ) > π (cid:16) (cid:16) α − (cid:17) − δ ( α ) (cid:17) x (log x ) (3.28)where now δ ( α ) = O (cid:16)(cid:16) α − (cid:17) c (cid:17) (3.29)for some c > .Recall that S f ( x, α ) ∼ π (cid:16) (cid:16) α − (cid:17)(cid:17) x (log x ) (3.30)corresponds to Hooley’s conjecture in the range < α ≤ . See the discussion in[F], § REMARK ON SOLUTIONS OF THE PELL EQUATION 13 R EFERENCES [B] J. Bourgain, The sum-product theorem in Z q with q arbitrary , J. Analyse Math, Vol. 106 (2008),1–93.[B-G1] J. Bourgain, M. Garaev, Sumsets of reciprocals in prime fields and multilinear Kloostermansums , to appear in Izvestia.[B-G2] J. Bourgain, M. Garaev, Kloosterman sums in residue rings , preprint 2013.[F] E. Fouvry, On the size of the fundamental solution of Pell equation , preprint.[H] C. Hooley, On the Pellian equation and the class number of indefinite binary quadratic forms ,Journal f¨ur die reine und angewandte Mathematik, Vol. 353, 98–131, 1984.S CHOOL OF M ATHEMATICS , I NSTITUTE FOR A DVANCED S TUDY , P RINCETON , NJ 08540 BOURGAIN @ MATH . IAS ..