aa r X i v : . [ m a t h . N T ] A ug ON QUADRATIC IRRATIONALS WITH BOUNDED PARTIALQUOTIENTS
J. BOURGAINA
BSTRACT . It is shown that for some explicit constants c > , A > , theasymptotic for the number of positive non-square discriminants D < x withfundamental solution ε D < x + α , < α < c , remains preserved if we requiremoreover Q ( √ D ) to contain an irrational with partial quotients bounded by A .
1. I
NTRODUCTION
We prove the following, related to McMullen’s conjecture on irrationals withbounded partial quotients in quadratic number fields.
Theorem 1.
For α < α < c ,there are atleast (cid:0) − o (1) (cid:1) α π √ x (log x ) positivenon-square integers D < x for which Q ( √ D ) contains an irrational which partialquotients are bounded by A and such that moreover the fundamental solution ε D tothe Pellequation t − Du = 1 isbounded by x + α .Here A, c > are explicit constant, and α > arbitrarily small and fixed. Wewill explain below the emphasis on the explicitness of these constants. Theorem 1certainly holds for α < . Remark 1.
According to the work of Hooley [H] and Fouvry [F], for < α ≤ , |{ ( ε D : D ); D ≤ x, D nonsquare, ε D ≤ D + α }| ∼ α π x (log x ) . (1.1)In [H], Hooley also conjectures asymptotics for α > of the form |{· · · }| ∼ B ( α ) x (log x ) (1.2)Thus in the range α < α < c , Theorem 1 recovers most of these discriminants. Remark 2.
Exploiting Theorem 1.12 in [M], the Theorem will follow from thefollowing statement.
Research supported in part by NSF Grants DMS 1301619. |{ D ≤ x ; D non-square and such that the Pell equation t − Du = 1 hasa solution with t < x + α and with the additional property that for some integer a < t, ( a, t ) = 1 , at has partial quotients bounded by A }|≥ (cid:0) − o (1) (cid:1) α π x (log x ) . (1.3)In order to prove (1.3), we combine Hooley’s approach recalled below with theresults from [B-K] around Zaremba’s conjecture. Denote Z = { t ∈ Z + ; there is < a < t, ( a, t ) = 1 such that at has partial quotients bounded by A } . (1.4)Observe that if t − Du = 1 and t < D , then necessarily t + u √ D is thefundamental solution ε D , since otherwise D > t > t + u √ D ≥ ε D ≥ D using the property ε D ≥ √ D . Hence, for < α ≤ , (1.3) will follow from thestatement X ( t,u,D ) ,t − Du =1 t ∈ Z,t 12 + α ,D 1+ 12 α 1+ 12 α The expression (1.7) was evaluated in [H] and [F]. We recall the argument.Since u < x α , clearly(1.7) ≥ X ≤ u ≤ X ρ ( u ) (cid:16) u √ x − u α u − O (1) (cid:17) > (cid:0) − o (1) (cid:1)n X ≤ u ≤ X ρ ( u ) u √ x − X ≤ u ≤ X ρ ( u ) u α − o . (1.9)Next, recalling [F], (24), (25) X u ≤ X ρ ( u ) u = 4 π (log X ) + O (log X ) (1.10) X u ≤ X ρ ( u ) u α − < CX α log X. (1.11)Substitution of (1.10), (1.11) leads to the required minoration (cid:0) − o (1) (cid:1) π √ x (log X ) = (cid:0) − o (1) (cid:1) α π √ x (log x ) . It remains to bound (1.8).We estimate (1.8) as follows(1.8) ≤ X ≤ u ≤ Xρ ( u ) > (log u ) ρ ( u ) √ xu (1.12) + X U ≤ X (log U ) X u ∼ U max ξ ( mod u ) |{ t ≤ U √ x ; t ≡ ξ ( mod u ) , t Z }| (1.13)where U takes dyadic values.We rely on the following statement, to be proven in the next section. Proposition 2. Thereis aconstant c > such that for U < y c X u ∼ U max ξ |{ t ≤ y ; t ≡ ξ ( mod u ) , t Z }| < yU (log y ) − (1.14)Estimate (1.12) by O ( √ x ) and, assuming α < c (1.15)and using the Proposition, also(1.13) . X U ≤ X (log U ) U √ xU (log x ) − . √ x. J. BOURGAIN This proves the Theorem. Remark 3. The proof of the Proposition appears in § c can be made explicit, since it relates to the ‘minor arcs analysis’ in [B-K] andnot to the spectral part. Note that |{ t < y ; t Z }| < y − δ for some δ > (see also [M-O-W]), so that obviously (1.24) holds with c < δ . The number δ depends however on the spectral theory of the continued fraction semigroup < a 11 0 ! ; 1 ≤ a ≤ A > + and is non-explicit.As mentioned above, our analysis permits to take c = in Theorem 1, but asit stands certainly does not cover the full range α ∈ ]0 , ] of Hooley’s result (1.1).2. P ROOF OF THE P ROPOSITION We use the analysis from [B-K] around the Zaremba problem. In particular,the constant A bounding the size of the partial quotients is taken at least A ≥ and may be further increased depending on our needs below. Let y = N be alarge integer. Following [B-K], replace Z ∩ [1 , N ] by a more convenient function θ : [1 , N ] → [0 , with the property that supp θ ⊂ Z . This function is introducedin [B-K] as the density on Z ∩ [1 , N ] obtained as a normalized image measure of asuitable subset Ω N of the semi-group S A under the map g 7→ h ge , e i . A major part of the analysis in [B-K] consisted then in the study of θ using theHardy-Littlewood circle method and exploiting the multilinear structure of Ω N .Some of this analysis will also be used here.We need to show that for U < N c X U ≤ u< U |{ n ≤ N ; n ≡ ξ u ( mod u ) , n Z }| < NU (log N ) − (2.1)for any assignments ξ u ( mod u ) .The condition n Z will be replaced by θ ( n ) = 0 . N QUADRATIC IRRATIONALS WITH BOUNDED PARTIAL QUOTIENTS 5 The major arcs analysis in the circle method leads to a singular series density(cf. [B-K], § σ ( n ) ≥ Y p | n (cid:16) p − (cid:17) Y p | n (cid:16) − p + 1 (cid:17) ≥ n . (2.2)Therefore it will suffice in order to establish (2.1) to show that X U ≤ u ≤ U X n ≡ ξ u ( mod u ) | θ ( n ) − σ ( n ) | . NU (log N ) . (2.3)Denote η = θ − σ and ˆ η ( α ) = P n η ( n ) e ( nα ) , α ∈ R / Z , its Fourier transform.By Parseval, X n ≡ ξ u ( mod u ) | η ( n ) | = 1 u Z (cid:12)(cid:12)(cid:12) X j ( mod u ) ˆ η (cid:16) ju + α (cid:17) e (cid:16) ξ u u j (cid:17)(cid:12)(cid:12)(cid:12) dα ≤ u Z | β | < u h X j ( mod u ) (cid:12)(cid:12)(cid:12) ˆ η (cid:16) ju + β (cid:17)(cid:12)(cid:12)(cid:12)i dβ (2.4)Define for dyadic KV q,K = { α ∈ T ; (cid:12)(cid:12)(cid:12) α − aq (cid:12)(cid:12)(cid:12) ∼ KN for some ( a, q ) = 1 } setting V q, = { α ∈ T ; (cid:12)(cid:12)(cid:12) α − aq (cid:12)(cid:12)(cid:12) . N for some ( a, q ) = 1 } . Since the function σ was obtained in [B-K] by restriction of ˆ θ to the major arcs set M = [ q ≤ Q K Denote ˆ θ = ˆ θ V c \M . (2.7)From the preceding, we may estimate (2.4) by U Z | β | < u h X j ( mod u ) (cid:12)(cid:12)(cid:12) ˆ θ (cid:16) ju + β (cid:17)(cid:12)(cid:12)(cid:12)i dβ (2.8) + X j ( mod u ) Z | β | < U (cid:12)(cid:12)(cid:12) ˆ θ (cid:16) ju + β (cid:17)(cid:12)(cid:12)(cid:12) T \ V c (cid:16) ju + β (cid:17) dβ. (2.9)By Lemma 1, (2.9) < N − c and the contribution in (2.3) at most U.N − c .Choose in (2.5) the constant c small enough to ensure the estimates below, bor-rowed from [B-K], valid. Then (2,1) will hold for c < c .In what follows, we always assume q, K < N c .In the sequel, we use the notation C for various numerical constants and let A be large enough to make C/A adequately small. Lemma 2. (see [B-K] , Prop, 5.2 and Prop. 6.3). | ˆ θ ( α ) | . N ( Kq ) − C/A if α ∈ V q,K . (2.10)The next statement is obtained by an easy variant of the proof of Prop. 6.21 in[B-K]. Lemma 3. Assume given for each q ∼ Q a (possibly empty) subset S q of theresidues ( mod q ) . Let | β | < KN . Then X q ∼ Q X a ∈ S q (cid:12)(cid:12)(cid:12) ˆ θ (cid:16) aq + β (cid:17)(cid:12)(cid:12)(cid:12) < ( KQ ) C/A Q (cid:16) X q | S q | (cid:17) N. (2.11)A duality argument permits then to derive from (2.11). Lemma 4. For | β | < KN , we have (cid:16) X q ∼ Q ( a,q )=1 (cid:12)(cid:12)(cid:12) ˆ θ (cid:16) aq + β (cid:17)(cid:12)(cid:12)(cid:12) (cid:17) < ( KQ ) C/A log QQ / N. (2.12) N QUADRATIC IRRATIONALS WITH BOUNDED PARTIAL QUOTIENTS 7 Hence Z ∪ q ∼ Q V q,K | ˆ θ ( α ) | < ( KQ ) C/A Q KN. (2.13)Let Λ ⊂ { q ∼ Q } . Combining (2.13) with (2.10) gives Lemma 5. For Λ as above, Q < N c X q ∈ Λ , ( a,q )=1 Z | β | < N − c (cid:12)(cid:12)(cid:12) ˆ θ (cid:16) aq + β (cid:17)(cid:12)(cid:12)(cid:12) dβ < | Λ | NQ − C/A . (2.14)We denote ( x, y ) (resp. [ x, y ] ) the ℓcd (resp. scm ) of x, y ∈ Z ∗ . Returning to(2.8) fix u ∼ U and assume j u + β ∈ V q,K with q, K < N c for some j ( mod u ) .Hence j u + β = a q + γ (2.15)where q < N c , ( a , q ) = 1 if a = 0 and | γ | < N − c . Then for all j ( mod u ) ju + β = j − j u + a q + γ ∈ V q ′ ,N c where q ′ | u q , u q < N c . In particular β = − j u + a q + γ = β + γ. (2.16)Denote ν p ( x ) the exponent of the prime p in the factorization of x ∈ Z ∗ . De-compose β as β = a q + a q + a q = a q + ˜ β (2.17)where ( a i , q i ) = 1 if a i = 0 and(2.18) q | u (2.19) If ν p ( q ) > , then ν p ( q ) > ν p ( u ) > and (cid:12)(cid:12) a q (cid:12)(cid:12) < q ,u ) (2.20) ( a, q ) = 1 .Note that since | β | ≤ | β | + | γ | < U + N − c < U and β − ˜ β ∈ u Z by(2.17), ˜ β essentially determines β . Next ju + β ∈ (cid:8) j ′ u ; j ′ ( mod u ) (cid:9) + ˜ β , hence ju + β = aq + ˜ β = a ′ q ′ where q | u , ( a, q ) = 1 = ( a ′ , q ′ ) . Note that ˜ β belongs tothe set S q of sums a q + a q where ( a i , q i ) = 1 if a i = 0 and(2.21) If ν p ( q ) > , then ν p ( q ) > ν p ( q ) and (cid:12)(cid:12) a q (cid:12)(cid:12) < q ,q ) (2.22) ( q , qq ) = 1 J. BOURGAIN as a consequence of (2.19), (2.20). With previous notation, we have for ˜ β ∈ S q that a ′ q ′ = aq + a q + a q = ˜ a [ q, q ] + a q with ˜ a = q ( q, q ) a + q ( q, q ) a , (˜ a, qq ) = 1 and q ′ = [ q, q ] q . We claim that a ′ q ′ essentially determines aq , ˜ β . Fix q , q , q divisors of q ′ . Since a ′ ≡ a [ q, q ]( mod q ) , ([ q, q ] q ) = 1 , a is determined andhence ˜ a . Recalling (2.21), | a | < q ( q,q ) and since q ( q,q ) a ≡ ˜ a (cid:0) mod q ( q,q ) (cid:1) , a isdetermined. This proves the claim.Estimate using Cauchy-Schwarz inequality X u ∼ U (2 . ≤ U X q q (cid:16) X q | u ,u ∼ U d ( u ) (cid:17) X ˜ β ∈ S q X ( a,q )=1 Z | γ | < N − c (cid:12)(cid:12)(cid:12) ˆ θ (cid:16) aq + ˜ β + γ (cid:17)(cid:12)(cid:12)(cid:12) dγ. (2 . If q | u , there is q | u such that q | q, q | q . We obtain (2 . ≤ U X q ≤ U d ( q ) X q | u,u ∼ U d (cid:16) uq (cid:17) X q | q,q | q q X ˜ β ∈ S q ( a,q )=1 Z | γ | < N − c | ˆ θ (cid:16) aq + ˜ β + γ (cid:17) | dγ . (log U ) U X q ≤ U d ( q ) q X q | qq | q q X q | q ′ ,q ′ Assuming A large enough and recalling (2.7) restricting Q ′ > Q , leads to theestimate (log U ) U Q − C/A < NU (log N ) − . (2 . This proves (2.1) and Proposition 2.R EFERENCES [B-K] J. Bourgain, A. Kontorovich, On Zaremba’s conjecture , to appear in Annals Math.[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 angewndte Mathematik, vol. 353, p98–131, 1984.[M] R. Mercat, Construction de fractions continues p´eriodiques uniform´ement born´ees , J. Th. Nom-bres Bordeaux, 25, no 1 (2013), 111–146.[M-O-W] M. Magee, H. Oh, D. Winter, Uniform congruence counting for Schottky semigroups in SL ( Z ) , arXiv:1601.03705v2.S CHOOL OF M ATHEMATICS , I NSTITUTE FOR A DVANCED S TUDY , P RINCETON , NJ 08540 BOURGAIN @ MATH . IAS .. . J. BOURGAIN