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Applications of Hyperbolic Geometry to Continued Fractions and Diophantine Approximation

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Applications of Hyperbolic Geometry to Continued Fractions and Diophantine Approximation Applications of hyperbolic geometry to continued fractions and Diophantine approximation Robert Hines University of Colorado, Boulder April 1, 2019 A picture Summary Our goal is to generalize features of the preceding picture to some nearby settings: H2 H3 (H2)r × (H3)s Hn P 1(R) P 1(C) P 1(R)r × P 1(C)s Sn−1 SL2(R) SL2(C) SL2(F ⊗ R) SVn−1(R) SL2(Z) SL2(O) " SV (O) ideal right-angled triangles ideal polyhedra horoball bounded geodesic " neighborhoods trajectories quad. quad./Herm. " " forms forms closed geodesics closed surfaces aniso. subgroups " Ingredients Ingredients Upper half-space models in dimensions two and three Hyperbolic two-space: 2 H = fz = x + iy 2 C : y > 0g; 2 1 @H = P (R); 2 Isom(H ) = P GL2(R); az + b az¯ + b g · z = ; (det g = ±1); cz + d cz¯ + d + ∼ Stab (i) = SO2(R)={±1g = SO2(R): Hyperbolic three-space: 3 H = fζ = z + jt 2 H : t > 0; z 2 Cg; 3 1 @H = P (C); 3 Isom(H ) = P SL2(C) o hτi; g · ζ = (aζ + b)(cζ + d)−1; τ(ζ) =z ¯ + jt; + ∼ Stab (j) = SU2(C)={±1g = SO3(R): Binary quadratic and Hermitian forms Hyperbolic two- and three-space are the Riemannian symmetric spaces associated to G = SL2(R), SL2(C). The points can be identified with roots of binary forms: 2 SL2(R)=SO2(R) ! fdet. 1 pos. def. bin. quadratic formsg ! H p −b + b2 − 4ac g 7! ggt = ax2 + bxy + cy2 = Q 7! =: Z(Q) 2a 3 SL2(C)=SU2(C) ! fdet. 1 pos. def. bin. Hermitian formsg ! H p −b + j ac − b¯b g 7! gg∗ = azz¯ + bzw¯ + ¯bzw¯ + cww¯ = H 7! =: Z(H) a We have Z(Qg) = g−1 · Z(Q);Z(Hg) = g−1 · Z(H); where the actions above are given by linear change of variable g and M¨obiustransformations g·. Binary quadratic and Hermitian forms (cont.) Indefinite forms parameterize codimension one geodesic subspaces: Q = ax2 + bxy + cy2 (b2 − 4ac > 0) p −b ± b2 − 4ac ∼ Z(Q) = 2a ∼ geodesic between the roots; H = azz¯ + bzw¯ + ¯bzw¯ + cww¯ (ac − jbj2 < 0) jbj2 − ac ∼ Z(H) = z : jz + b=aj2 = a2 ∼ geodesic plane with boundary Z(H): Once again the association is SL2-equivariant: Z(Qg) = g−1 · Z(Q);Z(Hg) = g−1 · Z(H): Geodesic and frame flows 2 We can identify SL2(R) and SL2(C) with spin bundles over H and 3 2 H , or P SL2(R) and P SL2(C) with the unit tangent bundle of H and the oriented orthonormal frame bundle of H3. Concretely (fixing base points (i; i) and (j; f1; i; jg) respectively and letting v 2 C, η 2 R + Ri + Rj be unit tangent vectors), we have the derivative action a b az + b v dg g · (z; v) = · (z; v) = ; = g · z; · v ; c d cz + d (cz + d)2 dz g · (ζ; η) = (g(ζ); (ζc + d)−1η(cζ + d)−1): In this setup, the geodesic flow and frame flow are given by et=2 0 t 7! g : 0 e−t=2 Some arithmetic quotients Given a number field F or signature (r; s) we have the R-algbra Q[x] Y R[x] Y R[x] F ⊗ R = ⊗ R = (m(x)) (x − σ(α)) (x2 − (σ(α) +σ ¯(α))x + σ(α)¯σ(α)) σ fσ;σ¯g real complex r s =∼ R × C ; and the ring of integers O is a discrete subring. The group Γ = SL2(O) ⊆ SL2(F ⊗ R) = G is a non-uniform lattice (non-cocompact, finite covolume discrete subgroup), and we will be interested in the symmetric and locally symmetric spaces 2 r 3 2 ∼ r s (H ) × (H ) = G=K; ΓnG=K (K = SO2(R) × SU2(C) ): For instance, when there is only one Archimedean place, we have the modular surface and the Bianchi orbifolds: SL2(Z)nSL2(R)=SO2(R); SL2(O)nSL2(C)=SU2(C) (where O is the ring of integers in an imaginary quadratic field). One can ask about approximating elements of F ⊗ R by elements of F . Mahler's compactness criterion n Given g 2 SLn(R), let Λg ⊆ R be the Z-span of the rows of g. (Oriented) change of basis corresponds to the coset SLn(Z)g. The basic compactness criterion in the space of unimodular lattices SLn(Z)nSLn(R) is the following. Mahler's criterion X ⊆ SLn(Z)nSLn(R) is precompact iff the lengths of all non-zero vectors in the corresponding unimodular lattices Λx; x 2 X are uniformly bounded below. Dani correspondence Badly approximable systems Bounded geodesic trajectories of linear forms () in the space of lattices For example: ξ 2 R is badly approximable, i.e. there exists C > 0 such that 2 jξ − p=qj ≥ C=q ; p=q 2 Q; if and only if 1 ξ e−t 0 SL ( ) SO ( ); t ≥ 0; 2 Z 0 1 0 et 2 R is bounded. Compactness of some orthogonal and unitary groups modulo Γ Example: [G = SL2(R), Γ = SL2(Z).] Let Q be an integral binay quadratic form and SO(Q; R) ⊆ G the group of g such that gtQg = Q. If Q(x; y) 6= 0 for all x; y 2 Z, then SO(Q; Z)nSO(Q; R) is compact in ΓnG. This follows from Mahler's criterion since k(x; y)gtk cannot be arbitrarily small for (0; 0) 6= (x; y) 2 Z2: x t x 1 ≤ (x; y)Q = (x; y)g Qg : y y Right-translating such quotients (i.e. changing basepoint) gives compact totally geodesic subspaces of ΓnG=K (closed geodesics on the modular surface in the example above). Vahlen group Generally speaking, Isom+(Hn) = O(1; n)◦, but we can cram this information into two-by-two matrices with entries in the definite Clifford algebra 2 R[e1; : : : ; en−2]; ei = −1; eiej = −ejei: We identify n−1 with the \paravectors" span f1; e ; : : : ; e g. The R R 1 n−2 orientation preserving isometries can be represented by the group a b SV = : ac∗; bd∗ 2 n−1; ad∗ − bc∗ = 1 : n c d R Here ∗ is the \reversal" involution induced by ∗ k(k−1)=2 (ei1 ··· eik ) = eik ··· ei1 = (−1) ei1 ··· eik : [When n = 2, 3, we get SL2(R) and SL2(C).] Prototype We now present some aspects of simple continued fractions, with an emphasis on badly approximable numbers (especially quadratic irrationals). Simple continued fractions: Definitions 1 The Euclidean algorthim [iterating (a; b) 7! (b; a mod b)] a = ba0 + r0; 0 ≤ r0 < b b = r0a1 + r1; 0 ≤ r1 < r0 r0 = r1a2 + r2; 0 ≤ r2 < r1 :::; or written in matrices a a 1 b = 0 b 1 0 r0 a 1 a 1 r = 0 1 0 1 0 1 0 r1 a 1 a 1 a 1 r = 0 1 2 1 1 0 1 0 1 0 r2 :::; expresses a rational number a=b as a finite continued fraction a 1 = a0 + 1 =: [a0; a1; : : : ; an]: b a1 + 1 a2+ ::: 1 an Simple continued fractions: Definitions 2 Extending this to irrational numbers ξ = bξc + fξg = a0 + ξ0 gives a dynamical system T : (0; 1) ! (0; 1); ξ0 7! f1/ξ0g and infinite sequences n+1 1 ξn+1 = f1/ξng = T ξ0; an+1 = b1/ξnc = n T ξ0 with 1 ξ = a0 + 1 =: [a0; a1; a2;:::]: a1 + a2+::: Stopping after n iterations gives rational approximations pn=qn to ξ, where p p a 1 a 1 a 1 n n−1 = 0 1 ::: n : qn qn−1 1 0 1 0 1 0 Simple continued fractions: Definitions 3 The branches of T −1 are all surjective and we have bijections ∼ R n Q = Z × NN; Q = f[a0; a1; : : : ; an]: n ≥ 0; an 6= 1 if n ≥ 1g: T is the left shift on these sequences, T ([0; a1; a2;:::]) = [0; a2; a3;:::]. 1.0 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0 Simple continued fractions: Approximation properties The convergents pn=qn to ξ have the following properties. Dirichlet bound 2 jξ − pn=qnj ≤ 1=qn Best approximations If 0 < q < qn then jqξ − pj > jqnξ − pnj; i.e. the continued fraction convergents are the best rational approximations to ξ. We say ξ is badly approximable if there exists C0 > 0 such that 0 2 jξ − p=qj ≥ C =q for all p; q 2 Z; i.e. the Dirichlet bound is tight (up to a multiplicative constant). This is an interesting class of real numbers (uncountable, measure zero, Hausdorff dimension 1, etc.) which is not completely understood. Characterizing badly approximable numbers: Bounded partial quotients One characterization of badly approximable numbers is the following. The number ξ = [a0; a1; a2;:::] 2 R n Q is badly approximable if and only its partial quotients an are bounded. [If ξ is badly approximable, jξ − p=qj ≥ C0=q2, then in particular C0 1 1 2 ≤ jξ − pn=qnj = 2 ≤ 2 ; qn qn([an+1; an+2;:::] + [0; an; : : : ; a1]) qnan+1 0 an+1 ≤ 1=C : Conversely, if the partial quotients are bounded, supnfang ≤ M, then for any p=q with 0 < q ≤ qn 1 jξ − p=qj ≥ jξ − pn=qnj = 2 qn(qn+1=qn + ξn+1) 1 1 1 = 2 ≥ 2 ≥ 2 ; qn([0; an+2;:::] + [an+1; an; : : : ; a1]) qn (an+1 + 2) qn(M + 2) using the fact that the convergents pn=qn are the best approximations.] Characterizing badly approximable numbers: Bounded geodesic trajectories Dani correspondence The number ξ is badly approximable if and only if the trajectory 1 ξ e−t 0 Ω = SL ( ) : t ≥ 0 ⊆ SL ( )nSL ( ) ξ 2 Z 0 1 0 et 2 Z 2 R is bounded (precompact).
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