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Dynamics of Units and Packing Constants of Ideals Perspectives • Continued Fractions in Q(√D) Dynamics of units and • The Diophantine semigroup packing constants of ideals • Geodesics in SL2(R)/SL2(Z) • (Classical) Arithmetic Chaos • Well-packed ideals Curtis T McMullen 1 Harvard University • Dynamics of units on P (Z/f) • Link Littlewood & Zaremba conjectures Continued Fractions Diophantine numbers Q. How to test if a real number x is in Q? x = [a , a , a , a , ...] = a + 1 0 1 2 3 0 a + 1 Q. How to test if a real number x is in Q(√D)? 1 a2 + 1 a3 + ... 1 x = [a0, a 1, a 2, a 3, ...] = a0 + a1 + 1 a2 + 1 BN = {x real : ai ≤N} a3 + ... BN-Q is a Cantor set of dim→1 as N→∞. A. x is in Q(√D) iff ai’s repeat. Conjecture: x algebraic and Diophantine iff x is rational or quadratic Diophantine sets in [0,1] Examples BN = {x : ai ≤N} γ = golden ratio = (1+√5)/2 = [1,1,1,1....] = [1] σ = silver ratio = 1+√2 = [2] [1,2,2,2] (√30) B2 [1,2] Q(√3) Q [1,2,2] Q(√85) [1,1,1,2] Q(√6) [1,1,2] Q(√10) [1,1,2,2] Q(√221) B4 Question: Does Q(√5) contain infinitely many periodic continued fractions with ai ≤ M? Thin group perspective Theorem Every real quadratic field contains infinitely GN = Diophantine semigroup in SL2(Z) many uniformly bounded, periodic continued fractions. generated by Wilson, Woods (1978) 01 01 01 ( 11) , ( 12) ,...( 1 N ) . Example: [1,4,2,3], [1,1,4,2,1,3], [1,1,1,4,2,1,1,3]... Theorem all lie in Q(√5). Infinitely many primitive A in GN have eigenvalues in Q(√D), provided N ≫ 0. Geometric perspective Thin group questions γ M = / SL2( ) GN = semigroup generated by ⊂ H Z 01 01 01 ( 11) , ( 12) ,...( 1 N ) . 3 x x´ Open Question ∈ Q(√D) Does {tr(A) : A in GN} have density one 2 ∞ in {1,2,3,4,...} for N ≫ 0? Theorem Given one loop of length L, there is a bounded a b Yes for ( c d )→d instead of (a+d). set B ⊂ M containing ∞ many loops with lengths in {L, 2L, 3L, 4L, ...}. Bourgain-Kontorovich Even though most loops of with these lengths are uniformly distributed (Duke). Example: Geodesic for Hyperbolic 3-manifolds [1,1,1,...,4,2,1,1,1...3] ∈ Q(√5) 3 γ ⊂ M = H / SL2([Z√-D]) beyond e.g. M = S3 - K continued fractions Theorem Again, there is a bounded set B ⊂ M made up of ∞ many loops with lengths in {L, 2L, 3L, 4L, ...}. Dynamical perspective Same scenario ⟹ chaotic dynamics Infinite orbit Finite orbit ... ... [1] [....1,1,4,2,1,1,1....] Cross-section to geodesic flow positive entropy Arithmetic chaos? Packing Perspective K/Q = number field of degree d Does the number of [a1,...,ap] in ∈ Q(√D) with ai ≤ 2 grow exponentially as the period p →∞? d J ⊂ K : an “ideal” in K (J ≃ Z ) Example: {ideals J} / K* = class ``group’’ (infinite) [1], [1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2], ...?... lie in Q(√5). Every J is an ideal for an order O(J) in K. N(J) Packing constant δ(J) = _____ Well-packed ideals det(J) Theorem `sphere’ N(x)<1 In any real quadratic field K=Q(√D), there are infinitely many ideal classes with δ(J) > δK > 0. Dictionary N(x) = Norm from K to Q = x1 ... xd x in K ⇔ J = Z + Zx N(J) = inf {|N(x)| : x in J, N(x) ≠0} 2 A in SL2(Z) ⇔ J = Z as a Z[A]-module det(J) = √|disc(J)| = √|det Tr(ai aj)| Unit group rank 1 Higher rank Conjecture [Cassel-Swinnerton-Dyer 1955 ] Conjecture In a total real cubic field K, only finitely many If K is a number field whose unit ideal classes satisfy δ(J) > δ > 0. group has rank one, then there are δ δ infinitely many ideal classes with (J) > K > 0. SL3( )/SL3( ) A·J R Z Cubic fields, 1 complex place Margulis Conjecture 2000 ⇒ Littlewood’s Conjecture & Quartic fields, 2 complex places Conjecture above No cubic case is known Einsiedler, Katok, Lindenstrauss, (e.g. x3 = x+1 is open) Michel, Venkatesh; CTM-, Minkowski’s Conjecture Fibonacci orders Fibonacci orders for golden mean D = the real quadratic order of discriminant D K = Q (√5) = real quadratic field O 2 2 ⋍ Z [x]/(x +bx+c), D = b -4c ε = unit in K = (1+√5)/2 K = Q (√D) = real quadratic field [ε] = D = 5 Z ∼ OD ε = unit in K 2 m D fm = discriminant of Z[ε ] D = discriminant of [ε] Z (f1,f2,...) = (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,...) 2 m D fm = discriminant of [ε ] Z m fm ~ ε , ε > 1 m th Z[ε ] = m Fibonacci order = f 2 D ∼ O m grows exponentially fast Class numbers Dynamics of units m 2 Acts on 2 Pic Z[ε ] has order about fm ~ √(D fm ) 01 Z U = (as large as possible) 11 Just as ε acts on Z[ε] Fibonacci Conjecture Acts on E = R2/Z2 and on E[ f ] ⋍ (Z/f )2 Given α>0, there is a δ>0 such that 2m m 1-α KEY FACT: U = Id on E[fm] |{J in Pic Z[ε ] : δ(J) > δ}| > fm for all m ≫ 0. Proof. m fm 1 fm 10 Lots of ideals ⇒ many well packed U = − fm+1 mod fm. fm fm+1 ≡ 01 (But not most) “Poincaré Recurrence” Dynamics of units (con’t) U2m = I 01 Acts on P1(Z/f) mod fm U = 11 {x = [a:b]} f192 = 597230427387774413556 9338397692020533504 J(x) = (a+b ε) + f [ε] an ideal for Df2 = 256 x .... Z Z O Pic 2 ≈ {orbits of U acting on 1( /f)} ODf P Z m m U = Id ⇒ Pic Z[ε ] has order ≈ fm / m large class numbers Height and packing The quadratic `field’ K = Q ! Q 1 01 Acts on P (Z/f) Z ! Z = quadratic ring 1 of discriminant 1 U = O Height H(x) = inf a2+b2 : x = [a:b] 11 2 Of = {(a,b) : a = b mod f} ⊂ Z ! Z H(x) = O(f) i δ( J(x)) ≈ mini H(U x)/f Conjecture Given α>0, there is a δ>0 s.t. i Events Ei = H(U x)/f > δ Each have probability p ≈ 1 1-α |{ J in Pic 2 : δ(J) > δ} > f Of Assume more or less independent! [Arithmetic chaos] for all f ≫ 0. m 1-α ⇒ |x : δ(J(x)) > δ| ≈ p fm ≥ fm [All class numbers large.] This conjecture implies Zaremba’s conjecture. ⇒ Fibonacci Conjecture Zaremba’s Conjecture Coda: Expanders from 2 Od ∃N : For any q>0, ∃ p/q = [a1,...,an] V(Xd) = {genus 2 covering spaces 2 2 with ai ≤ N. S → E = R /Z , branched over one point} How it follows: Pic f 2 ⋍ ( /f)* O Z J(x) = {(a,b) : a=xb mod f} ⊂ Z ! Z det J(x) = f SL2(Z) acts transitively ⇒ Tables built from 5 squares Graphs Xd δ( J(x))> δ ⇔ Conjecture: These Xd are expanders. N( J(x)) = minp,q {|q| |xq-pf|} > δf ⇔ |x/f - p/q| > δ/q2 ⇔ Xd = SL2(Z)/Γd, Γd not congruence! continued fraction of x/f is bounded by N(δ). Z*Z→Sd Jac(S) real multiplication by 2 Od Restrospective • CF’s; Geodesics/A-orbits; Packing constants • Arithmetic chaos in rank 1 • Margulis-Littlewood-CSD rigidity in higher rank Uniformly Diophantine numbers in a fixed real quadratic field. Compos. Math. 145(2009) slides online Main construction. We can now explicitly construct lattices with uni- formly bounded orbits under the action of U . ! " Theorem 2.2 Given A GL (Z) such that A2 = I, tr(A)=0and tr(A U)= ∈ 2 † 1,let ± m m Lm = U + U − A. Then for all m 0: ≥ 1. det(L ) = f is a generalized Fibonacci number; | m | 2m 2m 2. The lattice [Lm] is fixed by U ; 3. We have L m = LmA; − 4. For 0 i m we have: ≤ ≤ i i i i U LmU − , U − L mU C det Lm , (2.7) & & & − &≤ | | where C depends only on A and U. ! Proof. Our assumptions imply det(A)= 1. Since UU = I and U 2m = − † ± f2mU nf2m 1I,(2.5)gives − − m m m m det(Lm)=det(U )+det(U − A)+tr(U A†(U − )†) 2m = tr(A†U )= f ± ± 2m establishing (1). By construction Lm is integral, so Proposition 2.1 implies (2). Since A2 = I we have (3). For (4) first recall that f !i for i>0; in i ( particular, U i !i by (2.6). Thus for 0 i m we have & ± &≤ ≤ ≤ i i m i m i m U L U − = U + U − AU − = O(! )=O( f )=O( det L ). & m & & & 2m | m| i i AsimilarboundholdsforU L mU − ,whichgives(4).! ! − Corollary 2.3 There is a compact subset of PGL2(R)/ GL2(Z) which con- tains the lattices [U iL ] for all i, m Z. m ∈ Proof. Since A, U GL (Z)and[U 2mL ]=[L ], the lattices [U iL ]are ∈ 2 m m m represented in GL2(R)bythematrices i i i i U LmU − U − L mU and − det L det L | m| | m| with 0 i m.ThesematricesinturnlieinacompactsubsetofGL(R), ≤ ≤ ! ! 2 since they have determinant 1andtheirnormsareuniformlyboundedby ± (2.7). Projecting, we obtain a compact set in PGL2(R)/ GL2(Z)containing i the lattices [U Lm].
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