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We let µφ denote the normalized arc-length measure along φ, that is, for f ∈ Cc∞(X) we set 1 l(φ) µφ(f)= f(atx)dt, for some x ∈ φ, l(φ) ˆ0 where a denote the geodesic flow (see §2). For any I ⊆ G let µ := 1 µ t d d Id I φ Id φ | d| ∈ denote the normalized measure supported on Id and µd := µ d . Finally, given a sequence l( ) G P Gdk of subcollections I = {Idk } we let ϕ (k)= l(I ) . In this note, we prove the following: I dk
ϕI (k) Theorem 2. Let I = {Id } be a sequence of subcollections such that ψ(k) := k log(dk) tends to 0 as k →∞. Then, for any f ∈ Cc∞(X) we have
1 µI (f) − µX (f) ≤ C(f)ψ(k) 2 dk
where C is a constant depending only on f. In particular, µI equidistribute to µX . dk
Note that the only assumption on Id is about its total length. Theorem 2 answers a question raised in [11, Remark 6.1.]. In order to discuss a stronger variant of Theorem 2 and to put Theorem 2 in the context of remark [11, Remark 6.1.], one should contrast Theorem 2 with the results in [12, 8]. To explain these results, note that after choosing a base point, Gd inherits a structure from Pic(Od) (i.e. Gd is a Pic(Od)-torsor, see [5, §2] ). In [12, 8, 7], the authors establish the equidistribution of subcollections that correspond a 1 to subgroups of Hd < Pic(Od) with [Pic(Od) : Hd] ≫ d for some a < 2827 . In other words, they establish equidistribution of much smaller subcollections which are restricted by some "algebraic" condition. We note that these results, do not imply Theorem 2. First, in the context of Heegner points (which is the framework of the result in [8]), Theorem 2 is clearly false for arbitrary subcollections. Indeed, restricting to points which lie in a certain part of positive measure of the total space, yields subcollections Id with |Id|≥ C |Gd|, for some 0
It is important to note that a stronger, but non-effective, equidistribution result on 1 +o(1) subcollections follows from [5]. Indeed, the fact that l(Gd) = d 2 is the only infor- mation that is used in [5] to deduce that the limiting measure has maximal entropy and hence is equal to µX . Therefore, the same argument implies that any subcollections 1 +o(1) with l(Id)= d 2 also equidistribute (see mainly [5, Proposition 3.6]). Apart from the effectivity in the Theorem 2, we also remark that the following variant of Theorem 2 cannot be deduced from [5], i.e. using the above entropy argument. Our method uses as input an effective Duke’s Theorem, i.e., the effective equidistribution of γ Gd with d− savings for some γ > 0 (see §2.2). Note that by [12, 8, 7], a similar effective Theorem exists for collections Hd ⊂ Gd supported on cosets of subgroups Hd < Pic(Od) a 1 γ(a) with [Pic(Od) : Hd] ≫ d for some a < 2827 , with d− savings for some γ(a) > 0 (see [7, Corollary 1.4]). Therefore, with the exact same proof, it follows that Theorem 2 ϕ˜I (k) l( d ) ˜ H k holds for any subcollection I = {Idk ⊂Hdk } with ψ(k) := log(d ) where ϕ˜ (k)= l(I ) k I dk instead of ψ(k). This note is organized as follows: Theorem 2 is obtained by a simple application of effective mixing in conjunction with effective version of Duke’s Theorem. In hindsight, a similar argument is used in [13]. We review these ingredients in §2 and give the proof of Theorem 2 in §3. Section 4 is devoted for the construction of large but non- equidistributing subcollections as discussed above.
Acknowledgments. M.A. would like to thank Paul Nelson for fruitful conversations and patient explanation of the analytic methods behind effective statements of Duke’s Theorem and to Ilya Khayutin for many conversations. We also want to thank Elon Lindenstrauss for outlining the construction that appears in §4. M.A. acknowledges the support of ISEF and Advanced Research Grant 228304 from the ERC. While working on this project the authors visited the IIAS and its hospitality is great acknowledged.
2. Preliminaries t e 2 0 As above, let G = SL2(R), at := t , Γ be a finite-index congruence sub- 0 e− 2 ! 1 group of SL2(Z). Then, G acts on X = Γ \ G by g.Γx = Γxg− . Note a left invariant metric dG on G induces a metric on X which we denote by dX ([6, §9.3.2]). It is well 1 known (see e.g. [6, §9.4.2]) that under the identification of Γ \ G =∼ T (Y0(1)) the action C of at corresponds to the geodesic flow on X. We denote by Cc∞(X) ⊕ ⊂ C∞(X) the space of compactly supported smooth functions modulo the constants and by C0∞(X) := C f ∈ Cc∞(X) ⊕ : X fdµX = 0 . We define fT by ´ T 1 f (x) := f(a .x)dt. T T ˆ t 0 DUKE’STHEOREMFORSUBCOLLECTIONS 4
1 2 Finally, let 0 ≤ θ < 2 and assume that the unitary representation of G on L0(Γ \ G) does not weakly contain any complementary series with parameter ≥ θ (for Γ=SL2(Z) this holds with θ = 0, (the tempered case)).
2.1. Effective mixing. R Lemma 3. For any f ∈ C0∞(X), t ∈ ,ǫ> 0, we have 1 2 t((θ 2 )+ǫ) |hf, at ◦ fi| ≪ǫ S1(f) e − where S1 is a Sobolev norm. Proof. This assertion is proved in [13, Theorem §9.1.2] with an explicit Sobolev norm or 2 tβ in [9, page 216]. In fact, for our argument any bound of the form ≪ S1(f) e for some β < 0 will suffice.
Proposition 4. For any real valued f ∈ C0∞(X) we have S (f)2 µ |f |2 ≪ 1 X T T where S1 is the same as in Lemma 3. Proof. We have
2 2 1 µX |fT | = |fT | dµX = hfT ,fT i = hf,f ◦ at sidsdt = (∗). ˆ T 2 ˆ − X 0 t,s T ≤ ≤
Let B = {(t,s) : 0 ≤ t ≤ T, 0 ≤ s ≤ t} and note that since f is real-valued and at s◦ is − a unitary operator, we have hf,f ◦ at si = has t ◦ f,fi = hf,f ◦ as ti which implies that − − − T t 2 2 2 S1(f) β(t s) (∗)= hf,f ◦ at sidsdt ≤ |hf,f ◦ at si| dsdt ≪ e − dsdt T 2 ˆ − T 2 ˆ − T 2 ˆ ˆ B B 0 0
θ 1 − 2 where β := 2 < 0. A direct computation yields:
T t T 1 1 eβ(t s)dsdt = eβt − 1 dt = eβT − 1 − βT ˆ ˆ − β ˆ β2 0 0 0 and recalling that β < 0 we have reached the claim above: S (f)2 (∗) ≪ 1 . |β| T DUKE’STHEOREMFORSUBCOLLECTIONS 5
2.2. Effective Duke’s Theorem. The following theorem is now known as Duke’s The- orem [2]:
Theorem 5. There exist a γ > 0 and a sobolev norm S2, such that for any f ∈ Cc∞(X) we have γ |µd(f) − µX (f)|≤ S2(f)d− where S2(f) is the Sobolev norm on Cc∞(X) (whose further properties are discussed below). 2 As we want to apply Theorem 5 to |fT | , we need to bound the growth rate of 2 S2(|fT | ). To this end, note that C (1) There exists another sobolev norm S3 such that for any f1,f2 ∈ C0∞(X) ⊕ we have S2(f1f2) ≪ S3(f1)S3(f2). C κ (2) For any f ∈ C0∞(X) ⊕ and g ∈ G, S3(g.f) ≪ kgk S3(f) for some κ > 0, 1 1 where kgk denotes the operator norm Ad(g− ) : Lie(G) → Lie(G), X 7→ g− Xg. For the proof of Theorem 5 and the properties of the above Sobolev norms we refer the reader to [4, Theorem 4.6] and the references therein, in particular to [13, §2.9 and §6]. We thus have: C Lemma 6. There exists an α > 0 such that for any T > 0, and any f ∈ Cc∞(X) ⊕ we have 2 2 αT S2(|fT | ) ≪ S3(f) e . Proof. This readily follow from properties (1) and (2). Indeed, first use that that 2 S2(|fT | ) ≪ S3(fT )S3(fT ). Further, by the convexity of the norm S3 and Jensen’s inequality, we have T T 1 S3(f) κt κT S3(fT ) ≤ S3(f ◦ at)dt ≪ e dt ≪ S3(f)e , T ˆ0 T ˆ0 and the lemma follows.
3. Proof of Theorem 2 C For simplicity, we write Ik = Idk and µk = µIk . Fix f ∈ Cc∞(X)⊕ and set c = µX (f) and by abuse of notation, let c also denote the constant function c · 1X . As we aim to estimate |µk(f) − c|, note first that
µk(f) − c = µk(f − c)= µk ((f − c)T ) where the first equality follows since µk is a probability measure and the second since µk is supported on closed geodesics. By the Cauchy-Schwarz inequality, we have l(I ) 2 l(I ) 2 (3.1) k µ ((f − c) ) ≤ k µ 12 · µ |(f − c) |2 l(G ) k T l(G ) k X k T dk dk 2 l(Ik) 2 l(Ik) 2 (3.2) ≤ µk |(f − c)T | ≤ l µdk |(f − c)T | = (∗) ( dk ) l(Gdk ) G DUKE’STHEOREMFORSUBCOLLECTIONS 6
2 where the last inequality follows since the positivity of |(f − c)T | implies that
l(Ik) 2 2 µk |(f − c)T | ≤ µdk |(f − c)T | . l(Gdk ) 2 Now we apply Theorem 5. Note that |(f − c)T | does not have compact support, but it 2 2 is eventually constant since |(f − c)T | − c has compact support. Noting that µd and 2 µX are probability measures, we can apply Theorem 5 to estimate µdk |(f − c)T | and get that
l(Ik) 2 γ 2 2 (3.3) (∗) ≤ µX |(f − c)T | + dk− S2 |(f − c)T | − c = (∗∗) . l(Gdk ) 2 2 2 2 Note that S2 |(f − c)T | − c ≪ S2 |(f − c)T | + kfk . Now, as f − c has mean ∞ 2 zero we can apply Proposition 4 to estimate µX |(f − c)T | and Lemma 6 to bound 2 S2 |(f − c)T | , in order to get l(Ik) 2 1 2 γ αT γ 2 (∗∗) ≪f S1(f − c) T − + S3(f − c) dk− e + dk− kfk . l(Gdk ) ∞ Putting all of the above together and choosing T = η log(dk), we have (3.4) l(Ik) 2 2 1 1 2 ηα γ γ 2 (µk (f) − µX (f)) ≪f S1(f − c) η− log(dk)− + S3(f − c) dk − + dk− kfk . l(Gdk ) ∞
l( d ) ϕI (k) Choosing η < γ and multiplying both sides by ϕ (k)= G k , we get with ψ(k)= α I l(Ik) log(dk) that 2 (µk (f) − µX (f)) ≪f ψ(k) as claimed.
4. Large but non-equidistributing subcollections Z d+√d Recall that Od := [ 2 ], the unique order of discriminant d. The following con- struction was outlined to us by E. Lindenstrauss:
Theorem 7. Let {dk}k K ր∞ be any sequence with Reg(Odk ) ≪ log(dk). Given a> 0 ∈ 1 a 2 − and a fixed periodic orbit P , there exist subcollections Idk ⊂ Gdk with l(Idk ) ≫ dk such that any weak-* limit of a subsequence of µI gives a positive mass to P and in dk particular, the sequence µI does not equidistribute. dk k N n o ∈ Remark 8. Such sequences of discriminants do exist and even exist in any given fixed real quadratic field (see e.g. [10, §6 ]). Let P be a periodic orbit and note that since P is compact, it has a uniform injectivity radius which we denote by inj(P ). For any r< inj(P ) we let Ur = {x ∈ X : dX (x, P ) < r}. DUKE’STHEOREMFORSUBCOLLECTIONS 7
Lemma 9. For any small enough r > 0 and y ∈ Ur there exists an interval I of length ≍− log(r) such that for any t ∈ I, we have at.y ∈ U 1 . r 2
Proof. Let x ∈ P such that dX (x,y) < r for some r that will be determined momentarily. a b Denote x = Γg ,y = Γg and h = ∈ G with Γg h = Γg . Fix a norm on 1 2 c d 1 2 a b M2 2(R), say = max (|a| , |b| , |c| , |d|) and restrict it to G. We know that × c d the resulting metric d on G is bi-Lipschitz equivalent to dG (say κd ≤ dG ≤ Kd ), kk G kk X kk and for any z =Γ g0 ∈ P and r < min (inj(P ), 1) the projection Br (e) → Br (x), g 7→ r Γg0g is an isometry between dG and dX . Thus by assumption khk≤ κ and since at.y = 1 1 1 1 Γg1a−t athat− = at.x athat− we have dX (at.x, at.y) ≤ K athat− . As at.x ∈ P we have to show that there is an interval I of length ≍− log(r) such that t ∈ I implies 1 r 2 (4.1) a ha 1 ≤ . t t− K t 1 1 a be r r 2 Since athat− = t , and by assumption |a − 1| , |d − 1| , |b| , |c| ≤ κ ≪ K e− c d 1 t t r r 2 (where the last inequality holds for small enough r), (4.1) amounts to e |b|≤ e κ ≤ K 1 2 1 t t r r 2 and e− |c| ≤ e− κ ≤ K . Thus, for small enough r, we have dX (at.x, at.y) ≤ r if and only if 1 1 κr 2 κr 2 et ≤ − and e t ≤ − K − K κ 1 κ 1 if and only if t ≤ log( K ) − 2 log(r) and −t ≤ log( K ) − 2 log(r). As for small enough r, κ 1 κ 1 I = [− log( K )+ 2 log(r), log( K ) − 2 log(r)] has length ≍− log(r) we are done. Lemma 10. Let P be a fixed closed geodesic and 0 ≤ r ≤ min(1,inj(P )). There exists a function fr = f(r, P ) such that
(1) ∀x ∈ X, 0 ≤ fr(x) ≤ 1, (2) supp(fr) ⊂ Ur, fr|P ≡ 1 . b (3) S2(fr) ≪ r− for some b> 0 (where S2 is as in Theorem 5), c (4) µX (fr) := frdµX ≍ r for some c> 0. ´ Proof. As 0 ≤ r ≤ inj(P ), this construction takes place in the compact region of X. Therefore, to estimate the Sobolev norm any standard Sobolev norm on R3 will do. The most obvious construction works. Namely, note that the set U˜r = P · + {u (s)u−(s) : |s| < r} is a subset of Ur, and we can use the standard bump function 1 + + Θ(x) = exp( 2 ) on (−r, r) to define f : U˜ → R by f (Γg a u (s )u (s )) = (x−r) r r r P t 1 2 − Θ(s1)Θ(s2) where P = {ΓgP at}0 t l(P ) and |s2| , |s2| < r. One easily checks that fr has the desired properties. ≤ ≤ DUKE’STHEOREMFORSUBCOLLECTIONS 8
Proof of Theorem 7. For simplicity we restrain from mentioning injectivity radius issues any further as these may always be resolved by taking some variables to be large/small enough. Let γ be as in Theorem 5 and fix a> 0 and a periodic orbit P . Let η = η(a) > 0 that η will be determined later. Applying Theorem 5 to the functions fk := f(dk− , P ), which are provided by Lemma 10, we get that there exist C1,C2 > 0 such that γ cη (γ bη) µdk (fk) ≥ µX (fk) − dk− S2(fk) ≥ C1dk− − C2dk− − . γ Setting η so small such that γ − bη ≥ cη, i.e. η ≥ b+c , we have cη µdk (fk) ≫ dk− .
For a closed geodesic φ and f ∈ Cc∞(X) we set φ(f) to be the line integral of f along 1 φ. Recall that µdk (f) = l(Gdk )− φ φ(f), and note that since 0 ≤ fk ≤ 1, for ∈Gdk any φ ∈ G and any ǫ > 0, we have φ(f) ≤ Reg(O ) ≪ dǫ since by assumption dk P dk k Reg(Odk ) ≪ log(dk). Therefore, if we let Idk denote the subcollection of all the elements of Gdk that intersect the support of fk, for any ǫ> 0 we have
1 ( 2 +o(1) ǫ) cη 1 ǫ − − dk− ≪ µdk (fdk ) ≪ l(Gdk )− |Idk | dk = dk |Idk | .
1 1 ( 2 +o(1) ǫ) cη a − − 2 − Thus, by choosing ǫ(a) and η(a) accordingly, we have |Idk |≫ dk ≫ dk . Let ν be a weak-* limit of µI and we claim that ν(P ) > 0. It is enough to show dk that there exists a C > 0 such that for any large enough k, and any small enough r we k k have µI (Ur) >C. Let U = U − η and as d ր∞ it is clear that ∩ U = P . Thus it dk 2 k k dk k0 is enough to verify that for any k , µI (U ) >C for k ≫ 0. Fix k ∈ N; for any k ≥ k 0 dk 0 0 any element of φ ∈ Idk intersects supp(fk) by definition, and therefore there is a point η x ∈ φ with d(x, P ) ≤ dk− . By Lemma 9 there exists an interval Ik of length η log(dk) k k0 such that for any t ∈ Ik we have at.x ∈ φ ∩ U ⊂ φ ∩ U . Since the length of any element of Idk is Reg(Odk ) ≤ c1 log(dk) for some constant c1, any element of Idk spends η η at least c1 of its length in Uk0 . It follows that ν(Uk0 ) > c1 > 0 and so that ν(P ) > 0 and the claim follows.
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Section de mathématiques, EPFL, Station 8 - Bât. MA, CH-1015 Lausanne, Switzerland E-mail address: [email protected]
Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland E-mail address: [email protected]